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Littlewood Paley Theory and Square Functions

1 · Prerequisites

2 · Summary

This page develops the inhomogeneous Littlewood–Paley theory on Euclidean space: a fixed smooth dyadic frequency partition with a retained low-frequency block, the resulting operators, the square function, and the strict-range equivalence between the square-function norm and the $L^{p}$ norm for $1 < p < \infty$ . The analytic estimates assume Countable Choice through the cited measure and Fourier interfaces. The endpoint duality remark also inherits the full Axiom of Choice from its supplier.

The partition is built from a radial smooth cutoff $ψ$ equal to $1$ on the unit ball and supported inside the ball of radius $2$ ; the explicit construction vanishes at radius $3 / 2$ . The low piece is $φ_{0} = ψ$ ; for $j \geq 1$ set $φ_{j} = ψ (2^{- j} \cdot) - ψ (2^{- (j - 1)} \cdot)$ . All pieces are nonnegative, at most two of them are nonzero at any frequency, they sum to $1$ , their squares sum to a number in $[1 / 3, 1]$ , and their derivatives carry the scales $2^{- j ∣ α ∣}$ . The companion symbols ${\tilde{φ}}_{j} = φ_{j - 1} + φ_{j} + φ_{j + 1}$ reproduce the partition, $\sum_{j} {\tilde{φ}}_{j} φ_{j} = 1$ , and neither the low-frequency block $Δ_{0}$ nor its companion is assigned mean zero. The high-frequency kernels are rescalings of $K_{1}$ , the low-frequency kernel is $K_{0}$ , and the companion kernels are finite sums of neighbouring kernels. All have uniformly bounded $L^{1}$ mass, so every $Δ_{j}$ is uniformly bounded on $L^{p}$ for $1 \leq p \leq \infty$ .

The strict-range theorem is proved by Rademacher randomisation rather than by vector-valued Calderón–Zygmund theory. Khintchine's inequality for finite Rademacher sums—proved here from the equidistribution of finite Rademacher blocks, with sharp constants at $p = 2$ —converts the pointwise square function into an average of random signed sums, each of which is a Fourier multiplier with symbol $\sum_{j} \pm φ_{j}$ ; those symbols obey Mihlin's condition with constants independent of the signs, so the Mihlin multiplier theorem applies uniformly. The reproducing formula $f = \sum_{j} {\tilde{Δ}}_{j} Δ_{j} f$ in $S^{'}$ , its duality bound $\sum_{j} ∣ ⟨ Δ_{j} f, {\tilde{Δ}}_{j} g ⟩ ∣ \leq \int S f \tilde{S} g$ , and the $L^{p}$ norm-recovery corollary give the reverse inequality on Schwartz functions. Continuity of the finite truncations, monotone convergence and a Lipschitz estimate identify the extension to $L^{p}$ with the increasing pointwise square function. The same machinery gives the Littlewood–Paley characterisation of the Bessel-potential Hilbert–Sobolev spaces, $∥ f ∥_{H^{s}}^{2} ≍ \sum_{j} 2^{2 j s} ∥ Δ_{j} f ∥_{2}^{2}$ , and shows that the choice of admissible partition does not change the square-function space.

Only the strict range is claimed for the inhomogeneous square function. The endpoint remark identifies real Hardy space $H^{1}$ through its radial maximal-function definition and cites the proved local duality Real H1-BMO duality, under AC: bounded functionals on $H^{1}$ are represented by BMO functions modulo constants. This supplies no endpoint square-function characterization. The Lusin area function is defined as a conical functional, with no asserted equivalence to the square function. The cutoff must be smooth: sharp interval indicators have kernels of infinite $L^{1}$ norm, as recorded on the companion examples page.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Rademacher functions on the unit interval

Definition

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) for the Lebesgue-measure facts below. Let $I : = [0, 1)$ and let $λ$ be Lebesgue measure restricted to the Borel subsets of $I$ (Lebesgue measurable sets, the family $L (R^{n})$ , and the restricted set function $λ_{n}$ ). For integers $k \geq 1$ let $b_{k} (t) : = ⌊ 2^{k} t ⌋ - 2 ⌊ 2^{k - 1} t ⌋ \in {0, 1}$ be the $k$ -th binary digit, with $⌊ \cdot ⌋$ the integer part (Integer part: for every real $x$ there is exactly one integer $m$ with $m \leq x < m + 1$ ); the inclusion $b_{k} (t) \in {0, 1}$ follows from $⌊ 2 u ⌋ \in {2 ⌊ u ⌋, 2 ⌊ u ⌋ + 1}$ for $u = 2^{k - 1} t$ . For integers $j \geq 0$ define the $j$ -th Rademacher function $ε_{j} : I \to {\pm 1}$ by $ε_{j} (t) : = 1 - 2 b_{j + 1} (t) .$

Then:

Each $ε_{j}$ is Borel measurable (Borel measurable and Lebesgue measurable functions on $R^{n}$ ) and $∣ ε_{j} ∣ \equiv 1$ ; indeed $b_{k}$ is the parity of $⌊ 2^{k} t ⌋$ , so $ε_{j}$ is constant on the $2^{j + 1}$ half-open dyadic intervals of generation $j + 1$ , which are measurable (A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included), and the arithmetic of Arithmetic and lattice operations preserve measurability whenever they are defined preserves measurability.
$ε_{0} = 1$ on $[0, 1 / 2)$ and $- 1$ on $[1 / 2, 1)$ ; more generally, for every $j \geq 0$ the function $ε_{j}$ is constant, with alternating signs, on each half-open dyadic interval $[k 2^{- (j + 1)}, (k + 1) 2^{- (j + 1)})$ , $k = 0, \dots, 2^{j + 1} - 1$ , where it equals $(- 1)^{k}$ : on such an interval $2^{j + 1} t \in [k, k + 1)$ , so $⌊ 2^{j + 1} t ⌋ = k$ and $2^{j} t \in [k / 2, (k + 1) / 2)$ gives $⌊ 2^{j} t ⌋ = ⌊ k / 2 ⌋$ , hence $ε_{j} (t) = 1 - 2 (k - 2 ⌊ k / 2 ⌋) = (- 1)^{k}$ .
Consequently, since each half-open dyadic interval of generation $j + 1$ has Lebesgue measure $2^{- (j + 1)}$ (A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included) and the intervals partition $I$ , $\int_{0}^{1} ε_{j} (t) d t = \sum_{k = 0}^{2^{j + 1} - 1} (- 1)^{k} 2^{- (j + 1)} = 0 (j \geq 0),$ with the finite sum evaluated by pairing consecutive terms. Indeed, $ε_{j}^{+}$ and $ε_{j}^{-}$ are the indicators of the unions of the even and odd indexed intervals respectively, each of measure $1 / 2$ . Their nonnegative integrals equal $1 / 2$ (The integral of a nonnegative simple function, The nonnegative integral agrees with the simple integral on simple functions), so $ε_{j}$ is integrable and its signed integral is their difference (Integrable real and complex functions, and their integrals). The displayed identity is the $m = 1$ case of the equidistribution proved by the finite-block lemma below.

All integrals of functions of finitely many Rademacher functions on this page are integrals over $(I, λ)$ and are written $\int_{0}^{1}$ . The exponent $2^{j + 1}$ is an integer power in the sense of Integer powers $a^{m}$ . No choice principle is used to construct the binary digits or sign functions; the measure and integral assertions inherit Countable Choice from the cited Lebesgue-measure suppliers.

LemmaStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Finite Rademacher blocks are equidistributed

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $j_{1} < \dots < j_{m}$ be nonnegative integers and let $F : {\pm 1}^{m} \to C$ be any function. Then $\int_{0}^{1} F (ε_{j_{1}} (t), \dots, ε_{j_{m}} (t)) d t = 2^{- m} \sum_{s \in {\pm 1}^{m}} F (s) .$ Consequently, for all indices $l_{1}, \dots, l_{N} \geq 0$ one has $\int_{0}^{1} \prod_{i = 1}^{N} ε_{l_{i}} (t) d t = 1$ when every index $l$ occurs an even number of times and $0$ otherwise; in particular $\int_{0}^{1} ε_{j} d t = 0$ and $\int_{0}^{1} ε_{j} ε_{k} d t = δ_{j k}$ for all $j, k \geq 0$ .

Facts & Assumptions

Given: Countable Choice and the Rademacher functions $ε_{j}$ of Rademacher functions on the unit interval, integers $m \geq 1$ , $0 \leq j_{1} < \dots < j_{m}$ , and a function $F : {\pm 1}^{m} \to C$ . Integrals over $I = [0, 1)$ are written $\int_{0}^{1}$ ; $J : = j_{m} + 1$ and $d_{i} : = j_{m} - j_{i}$ for $i = 1, \dots, m$ .

[F1]

For every $j \geq 0$ the function $ε_{j}$ is Borel measurable, $∣ ε_{j} ∣ \equiv 1$ , and it is constant on each half-open dyadic interval $[k 2^{- (j + 1)}, (k + 1) 2^{- (j + 1)})$ , $k = 0, \dots, 2^{j + 1} - 1$ , where it equals $(- 1)^{k}$ (Rademacher functions on the unit interval).

[F2]

Every half-open box in $R^{n}$ is Lebesgue measurable with measure equal to the product of its side lengths, in particular $λ ([a, b)) = b - a$ on $R$ (A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included); the integral over a measurable set and the fact that almost everywhere equal functions have equal integrals are the conventions of Integral over a measurable subset and Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree.

[F3]

A nonnegative simple measurable function has nonnegative integral equal to its simple integral, and the complex integral is $\int h = \int Re h + i \int Im h$ (A simple function and its canonical representation, The integral of a nonnegative simple function, The nonnegative integral agrees with the simple integral on simple functions, Integrable real and complex functions, and their integrals).

[F4]

Finite sums are the recursion of Finite sums and finite products, by recursion and satisfy additivity, scaling, splitting and monotonicity (Laws of finite sums and finite products).

[F5]

Applying real arithmetic closure to real and imaginary parts shows that sums and products of measurable complex functions are measurable (Arithmetic and lattice operations preserve measurability whenever they are defined).

Proof

technique · direct

1.1givenF4algebra

Binary counting. Set $J : = j_{m} + 1 \geq 1$ and $d_{i} : = j_{m} - j_{i}$ , so $0 = d_{m} < d_{m - 1} < \dots < d_{1} < J$ . Successive division by $2$ gives a unique remainder in ${0, 1}$ and a quotient less than $2^{J - 1}$ ; induction on $J$ , starting with $J = 0$ and $k = 0$ , therefore gives every integer $k$ with $0 \leq k < 2^{J}$ a unique binary expansion $k = \sum_{r = 0}^{J - 1} β_{r} (k) 2^{r}$ with digits $β_{r} (k) \in {0, 1}$ . For every integer $d$ with $0 \leq d < J$ one has $⌊ k / 2^{d} ⌋ = \sum_{r = d}^{J - 1} β_{r} (k) 2^{r - d}$ , whose parity is $β_{d} (k)$ because all terms with $r > d$ are even multiples of $2$ ; hence $(- 1)^{⌊ k / 2^{d} ⌋} = (- 1)^{β_{d} (k)}$ . The map $k \mapsto (β_{d_{1}} (k), \dots, β_{d_{m}} (k))$ from ${0, \dots, 2^{J} - 1}$ onto ${0, 1}^{m}$ has every fiber of cardinality exactly $2^{J - m}$ : for prescribed digits $b_{1}, \dots, b_{m}$ the solutions are precisely $k = \sum_{i = 1}^{m} b_{i} 2^{d_{i}} + \sum_{r \notin {d_{1}, \dots, d_{m}}, r < J} c_{r} 2^{r}$ with $c_{r} \in {0, 1}$ , and distinct choices of the free digits $c_{r}$ give distinct $k$ by uniqueness of the binary expansion, while there are $J - m$ free digits.

2.1F1step 1.1algebra

Constant values on the dyadic intervals. Let $I_{k} : = [k 2^{- J}, (k + 1) 2^{- J})$ for $k = 0, \dots, 2^{J} - 1$ ; these intervals partition $I$ exactly. If $t \in I_{k}$ then $2^{j_{i} + 1} t \in [k / 2^{d_{i}}, (k + 1) / 2^{d_{i}}) \subset [⌊ k / 2^{d_{i}} ⌋, ⌊ k / 2^{d_{i}} ⌋ + 1)$ , because $k = 2^{d_{i}} ⌊ k / 2^{d_{i}} ⌋ + r$ with $0 \leq r < 2^{d_{i}}$ gives $(k + 1) / 2^{d_{i}} = ⌊ k / 2^{d_{i}} ⌋ + (r + 1) / 2^{d_{i}} \leq ⌊ k / 2^{d_{i}} ⌋ + 1$ ; hence $⌊ 2^{j_{i} + 1} t ⌋ = ⌊ k / 2^{d_{i}} ⌋$ and, by [F1], $ε_{j_{i}} (t) = (- 1)^{⌊ k / 2^{d_{i}} ⌋} = (- 1)^{β_{d_{i}} (k)}$ . Therefore the sign vector $(ε_{j_{1}} (t), \dots, ε_{j_{m}} (t))$ equals $s (k) : = ((- 1)^{β_{d_{1}} (k)}, \dots, (- 1)^{β_{d_{m}} (k)})$ on all of $I_{k}$ .

3.1F2step 1.1step 2.1algebra

The level sets have measure $2^{- m}$ . For a sign pattern $s \in {\pm 1}^{m}$ put $E_{s} : = {t \in I : (ε_{j_{1}} (t), \dots, ε_{j_{m}} (t)) = s}$ and let $N_{s} : = # {k < 2^{J} : s (k) = s}$ . By step 2.1 the set $E_{s}$ equals the union of the intervals $I_{k}$ over those $k$ ; the intervals are pairwise disjoint, each has measure $2^{- J}$ by [F2], and the defining map $s (\cdot)$ is a bijection between sign patterns and digit vectors, so by step 1.1 $N_{s} = 2^{J - m}$ ; hence $λ (E_{s}) = N_{s} 2^{- J} = 2^{- m}$ .

4.1F3F5step 3.1algebra

The identity for $F \geq 0$ . Suppose first that $F \geq 0$ . The composition $F \circ Φ$ , $Φ : = (ε_{j_{1}}, \dots, ε_{j_{m}})$ , is a nonnegative simple measurable function constant on the finitely many measurable sets $E_{s}$ : it equals $F (s)$ on $E_{s}$ , the union of the $E_{s}$ is all of $I$ , and measurability follows from [F5]. Hence, by [F3], $\int_{0}^{1} F \circ Φ d λ$ equals its simple integral $\sum_{y \geq 0} y λ ({F \circ Φ = y}) = \sum_{s} F (s) λ (E_{s}) = 2^{- m} \sum_{s} F (s)$ by step 3.1, where the last sum groups the $s$ with equal value $F (s)$ . For a real-valued $F$ write $F = F^{+} - F^{-}$ ; then $(F \circ Φ)^{\pm} = F^{\pm} \circ Φ$ and the real integral is the difference of the two nonnegative integrals, so the identity holds; for complex-valued $F$ apply this to $Re F$ and $Im F$ and combine with the definition of the complex integral in [F3].

5.1step 4.1F4algebra∎

The monomial and orthonormality formulas. If $N = 0$ , the empty product is $1$ and its integral is $1$ . For $N \geq 1$ , let $l_{1}, \dots, l_{N} \geq 0$ have distinct values $j_{1} < \dots < j_{m}$ with multiplicities $e_{1}, \dots, e_{m} \geq 1$ . Apply step 4.1 to the function $F (s_{1}, \dots, s_{m}) : = \prod_{i = 1}^{m} s_{i}^{e_{i}}$ : since $s^{e} = 1$ for even $e$ and $s^{e} = s$ for odd $e$ , the average factorises as $2^{- m} \sum_{s} \prod_{i} s_{i}^{e_{i}} = \prod_{i = 1}^{m} (\frac{1}{2} \sum_{u = \pm 1} u^{e_{i}})$ , and each factor is $1$ for even $e_{i}$ and $\frac{1}{2} (1 - 1) = 0$ for odd $e_{i}$ . Hence the integral is $1$ when every index occurs an even number of times and $0$ otherwise. Taking $m = 1$ gives $\int_{0}^{1} ε_{j} = 0$ ; taking $N = 2$ gives $\int_{0}^{1} ε_{j} ε_{k} = 1$ for $j = k$ and $= 0$ for $j \neq k$ , that is $δ_{j k}$ .

TheoremStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Khintchine's inequality for finite Rademacher sums

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). For every finite sequence $(a_{j})_{j \in J}$ of complex numbers, indexed by a finite set $J \subset {0, 1, 2, \dots}$ , and every $0 < p < \infty$ there are constants $0 < c_{p} \leq C_{p} < \infty$ , depending only on $p$ , such that $c_{p} (\sum_{j \in J} ∣ a_{j} ∣^{2})^{1 / 2} \leq (\int_{0}^{1} ∣ \sum_{j \in J} ε_{j} (t) a_{j} ∣^{p} d t)^{1 / p} \leq C_{p} (\sum_{j \in J} ∣ a_{j} ∣^{2})^{1 / 2},$ and the constants do not depend on the finite set $J$ . For $p = 2$ both inequalities hold with constants $1$ : $\int_{0}^{1} ∣ \sum_{j \in J} ε_{j} (t) a_{j} ∣^{2} d t = \sum_{j \in J} ∣ a_{j} ∣^{2} .$ The empty sum is zero and the empty case is trivial.

Facts & Assumptions

Given: Countable Choice, a nonempty finite set $J \subset {0, 1, 2, \dots}$ , complex numbers $a_{j}$ $(j \in J)$ , and $0 < p < \infty$ ; write $S (t) : = \sum_{j \in J} ε_{j} (t) a_{j} = A (t) + i B (t)$ with $A (t) = \sum_{j} ε_{j} (t) Re a_{j}$ and $B (t) = \sum_{j} ε_{j} (t) Im a_{j}$ , and $s^{2} : = \sum_{j} ∣ a_{j} ∣^{2}$ , $u^{2} : = \sum_{j} (Re a_{j})^{2}$ , $v^{2} : = \sum_{j} (Im a_{j})^{2}$ , so $u^{2} + v^{2} = s^{2}$ .

[F1]

For a nonempty finite set $J$ of nonnegative indices and any function $F : {\pm 1}^{J} \to C$ one has $\int_{0}^{1} F (ε_{j} (t) : j \in J) d t = 2^{- ∣ J ∣} \sum_{s \in {\pm 1}^{J}} F (s)$ ; consequently $\int_{0}^{1} ε_{j} d t = 0$ and $\int_{0}^{1} ε_{j} ε_{k} d t = δ_{j k}$ (Finite Rademacher blocks are equidistributed).

[F2]

$\exp (x) = \sum_{n \geq 0} x^{n} / n!$ for real $x$ and $\cosh y = (e^{y} + e^{- y}) / 2$ for real $y$ ; moreover $\cosh y = \sum_{k \geq 0} y^{2 k} / (2 k)!$ and $\cosh y \geq 0$ (The real exponential function and the number $e$ by a power series, The six hyperbolic functions and their natural domains). The addition law $e^{x + y} = e^{x} e^{y}$ , positivity $e^{x} > 0$ and strict increase follow from The exponential addition formula $\exp (x + y) = \exp (x) \exp (y)$ , The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ and The exponential function is strictly increasing.

[F3]

For every real $y$ one has $0 \leq \cosh y \leq e^{y^{2} / 2}$ : since $(2 k)! = 1 \cdot 2 \dots 2 k \geq 2^{k} k!$ for every $k$ (pairing $2 j - 1, 2 j$ ), the nonnegative series $\sum_{k} y^{2 k} / (2 k)!$ is termwise dominated by $\sum_{k} y^{2 k} / (2^{k} k!) = e^{y^{2} / 2}$ .

[F4]

The nonnegative Lebesgue integral is monotone and scales constants: if $0 \leq f \leq g$ then $\int f \leq \int g$ , and $\int c 1_{E} = c λ (E)$ for $c \geq 0$ (Monotonicity and nonnegative homogeneity of the nonnegative integral); the layer-cake formula $\int ∣ h ∣^{p} = p \int_{0}^{\infty} λ (∣ h ∣ > t) t^{p - 1} d t$ holds for measurable complex $h$ and $0 < p < \infty$ (For 0 < p < infinity, the layer-cake formula computes the integral of |f|^p from the distribution function).

[F5]

Holder's inequality for $L^{p}$ and $L^{q}$ with $1 / p + 1 / q = 1$ , in particular Cauchy-Schwarz, and the triangle inequality for integrals (Complex Holder, Minkowski, and the quotient norm, The modulus of an integral is bounded by the integral of the modulus).

[F6]

[F7]

For every $s > 0$ the Euler integral $Γ (s) = \int_{0}^{\infty} t^{s - 1} e^{- t} d t$ converges, so it is a finite positive number (The real Gamma function by Euler's integral, Euler's Gamma integral converges exactly for positive real parameters). The monotone $C^{1}$ substitution $t = 2 s \sqrt{τ}$ on $(0, \infty)$ is valid on compact truncations and at both improper ends by Change of variable in an improper integral.

Proof

technique · direct

1.1F1F6algebra

Setup and second moment. The functions $S, A, B$ are finite sums of products of constants with the measurable functions $ε_{j}$ , hence measurable by [F6], and $∣ S ∣^{2} = A^{2} + B^{2}$ . By [F1] the mean of $S$ vanishes, $\int_{0}^{1} S d t = 0$ , and expanding $∣ S ∣^{2} = \sum_{j, k} ε_{j} ε_{k} a_{j} {\overset{ˉ}{a}}_{k}$ and integrating termwise with $\int ε_{j} ε_{k} = δ_{j k}$ gives $\int_{0}^{1} ∣ S ∣^{2} d t = \sum_{j} ∣ a_{j} ∣^{2} = s^{2}$ .

2.1F1F2F3step 1.1algebra

Exponential moments. For real $ρ$ the function $t \mapsto e^{ρ A (t)}$ is a finite product $\prod_{j \in J} e^{ρ ε_{j} (t) Re a_{j}}$ of functions of the individual signs, so [F1] applied to $F (s) = \prod_{j} e^{ρ s_{j} Re a_{j}}$ gives $\int_{0}^{1} e^{ρ A} d t = \prod_{j \in J} \frac{e^{ρ Re a_{j}} + e^{- ρ Re a_{j}}}{2} = \prod_{j} \cosh (ρ Re a_{j}) \leq \prod_{j} e^{(ρ Re a_{j})^{2} / 2} = e^{ρ^{2} u^{2} / 2}$ , where [F2] and [F3] were used termwise and the product of exponentials was combined.

3.1F2F4step 2.1algebra

Tail bounds. Let $λ > 0$ . If $u > 0$ , take $ρ = λ / u^{2}$ in step 2.1 and use monotonicity of the integral on ${A > λ}$ to get $λ ({A > λ}) \leq e^{- ρ λ} \int e^{ρ A} \leq e^{- λ^{2} / (2 u^{2})}$ ; applying the same argument to $- A$ gives $λ ({∣ A ∣ > λ}) \leq 2 e^{- λ^{2} / (2 u^{2})}$ . If $u = 0$ , then $A = 0$ and this tail measure is $0$ . The same bounds hold for $B$ with $v$ . If $s = 0$ , then $S = 0$ and the tail measure is $0$ . Otherwise $s > 0$ , and ${∣ S ∣ > λ} \subset {∣ A ∣ > λ / \sqrt{2}} \cup {∣ B ∣ > λ / \sqrt{2}}$ because if both component moduli are at most $λ / \sqrt{2}$ , then $∣ S ∣^{2} = A^{2} + B^{2} \leq λ^{2}$ . Applying the component bound at threshold $λ / \sqrt{2}$ separately to the labeled $A$ - and $B$ -tails, and omitting a contribution when its variance is zero, gives $λ ({∣ S ∣ > λ}) \leq {\begin{cases} 2 e^{- λ^{2} / (4 u^{2})}, & u > 0, \\ 0, & u = 0 \end{cases} + {\begin{cases} 2 e^{- λ^{2} / (4 v^{2})}, & v > 0, \\ 0, & v = 0 \end{cases} \leq 4 e^{- λ^{2} / (4 s^{2})},$ because each positive variance among $u, v$ is at most $s$ and the two labeled contributions are each at most $2 e^{- λ^{2} / (4 s^{2})}$ . Equal positive variances still contribute twice, as required by the union bound.

4.1F4F7step 3.1algebra

Upper bound. For $s > 0$ the layer-cake formula [F4] applied to $S$ and step 3.1 give $\int_{0}^{1} ∣ S ∣^{p} d t = p \int_{0}^{\infty} λ ({∣ S ∣ > t}) t^{p - 1} d t \leq 4 p \int_{0}^{\infty} e^{- t^{2} / (4 s^{2})} t^{p - 1} d t$ ; substituting $t = 2 s \sqrt{τ}$ , $d t = s τ^{- 1 / 2} d τ$ , turns the last integral into $4 p 2^{p - 1} s^{p} \int_{0}^{\infty} τ^{p / 2 - 1} e^{- τ} d τ = 4 p 2^{p - 1} Γ (p / 2) s^{p}$ , which is finite by [F7]. Hence $∥ S ∥_{p} \leq C_{p} s$ with $C_{p} : = (4 p 2^{p - 1} Γ (p / 2))^{1 / p} < \infty$ and, for $s = 0$ , $∥ S ∥_{p} = 0$ .

5.1F5step 1.1step 4.1algebra

Lower bound. Let $s > 0$ . Step 4.1 with $p = 4$ gives $\int_{0}^{1} ∣ S ∣^{4} d t \leq C_{4}^{4} s^{4}$ ; splitting the integral of $∣ S ∣^{2}$ over ${∣ S ∣ \leq s / 2}$ and ${∣ S ∣ > s / 2}$ and applying Cauchy-Schwarz [F5] to the second piece, $\int_{0}^{1} ∣ S ∣^{2} d t \leq (s / 2)^{2} + (\int ∣ S ∣^{4})^{1 / 2} λ ({∣ S ∣ > s / 2})^{1 / 2} \leq s^{2} / 4 + C_{4}^{2} s^{2} λ ({∣ S ∣ > s / 2})^{1 / 2}$ ; with $\int ∣ S ∣^{2} = s^{2}$ from step 1.1 this gives $λ ({∣ S ∣ > s / 2}) \geq (3 / (4 C_{4}^{2}))^{2} = 9 / (16 C_{4}^{4})$ . Consequently, for every $p > 0$ , $\int_{0}^{1} ∣ S ∣^{p} d t \geq \int_{{∣ S ∣ > s / 2}} ∣ S ∣^{p} d t \geq (s / 2)^{p} \cdot 9 / (16 C_{4}^{4}) = c_{p}^{p} s^{p}$ with $c_{p} : = (9 / (16 C_{4}^{4}))^{1 / p} / 2 > 0$ .

6.1step 1.1step 4.1step 5.1algebra∎

Conclusion. For a nonempty $J$ and $s > 0$ , steps 4.1 and 5.1 give the two-sided inequality with constants $c_{p}, C_{p}$ that are explicit functions of $p$ alone, in particular independent of $J$ and of the coefficients; for $s = 0$ all quantities vanish and the inequality is trivial, and for the empty set $J = \emptyset$ both sides are $0$ . The case $p = 2$ is step 1.1, where the identity $\int ∣ S ∣^{2} = s^{2}$ gives both inequalities with constants $1$ .

LemmaStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Existence of a smooth inhomogeneous dyadic frequency partition

Statement

For every $n \geq 1$ there is a radial function $ψ \in C_{c}^{\infty} (R^{n})$ with $0 \leq ψ \leq 1$ , $ψ (ξ) = 1$ for $∣ ξ ∣ \leq 1$ , $ψ (ξ) = 0$ for $∣ ξ ∣ \geq 3 / 2$ , and $supp ψ \subset {∣ ξ ∣ < 2}$ . For any radial $ψ \in C_{c}^{\infty} (R^{n})$ with $0 \leq ψ \leq 1$ , $ψ = 1$ on $∣ ξ ∣ \leq 1$ and $supp ψ \subset {∣ ξ ∣ < 2}$ set $φ_{0} : = ψ$ and $φ_{j} (ξ) : = ψ (2^{- j} ξ) - ψ (2^{- (j - 1)} ξ)$ for $j \geq 1$ . Then each $φ_{j}$ is radial, real-valued, lies in $C_{c}^{\infty} (R^{n})$ , and:

$\sum_{j \geq 0} φ_{j} (ξ) = 1$ for every $ξ$ , the sum being locally finite;
$supp φ_{0} \subset {∣ ξ ∣ \leq 2}$ and, for $j \geq 1$ , $supp φ_{j} \subset {2^{j - 1} \leq ∣ ξ ∣ \leq 2^{j + 1}}$ ;
$φ_{j} \geq 0$ for every $j$ , at every $ξ$ at most three of the functions $φ_{j}$ are nonzero, and $\frac{1}{3} \leq \sum_{j \geq 0} φ_{j} (ξ)^{2} \leq 1$ ;
for every multi-index $α$ there is $C_{α} = C_{α} (n, ψ) < \infty$ with $∣ \partial^{α} φ_{j} (ξ) ∣ \leq C_{α} 2^{- j ∣ α ∣}$ for $j \geq 1$ and all $ξ$ , and $∣ \partial^{α} φ_{0} (ξ) ∣ \leq C_{α}$ for all $ξ$ ; consequently $∣ \partial^{α} φ_{j} (ξ) ∣ \leq 2^{∣ α ∣} C_{α} ∣ ξ ∣^{- ∣ α ∣}$ for $j \geq 1$ and $ξ \neq 0$ .

Facts & Assumptions

Given: an integer $n \geq 1$ , Gaussian brackets and multi-indices as in $C^{k}$ maps and multi-index derivative notation in Euclidean space, and the support convention of The support of a function on $R^{n}$ and its compactly supported Riemann integral. Write $∥ f ∥_{\infty} : = \sup_{ξ} ∣ f (ξ) ∣$ for bounded functions.

[F1]

The standard smooth step $σ (t) : = β (t) / (β (t) + β (1 - t))$ , with $β$ the standard flat function, satisfies $σ \in C^{\infty} (R)$ , $0 \leq σ \leq 1$ , $σ (t) = 0$ for $t \leq 0$ and $σ (t) = 1$ for $t \geq 1$ (The standard smooth step function); $β$ is smooth on $R$ (The standard flat function is smooth and flat at zero).

[F2]

If $q : R^{n} \to R$ is smooth and $σ : R \to R$ is smooth, then $σ \circ q$ is smooth: the chain rule for total derivatives gives the first derivative and iteration gives all higher ones (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , $C^{k}$ Euclidean maps and diffeomorphisms).

[F3]

$C_{c}^{\infty} (R^{n}) \subset S (R^{n})$ : a compactly supported smooth function has all its derivatives bounded, hence finite seminorms (Schwartz space and its seminorms).

[F4]

For $h \in C^{\infty} (R^{n})$ and $R > 0$ the chain rule (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ) gives $\partial_{i} (h (\cdot / R)) (ξ) = R^{- 1} (\partial_{i} h) (ξ / R)$ . Iterating this identity in the prescribed multi-index order gives $\partial^{β} (h (\cdot / R)) (ξ) = R^{- ∣ β ∣} (\partial^{β} h) (ξ / R)$ , with $β = 0$ the identity itself.

Proof

technique · direct

1.1F1F2F3algebra

Construction. Put $q (ξ) : = (9 - 4 ∣ ξ ∣^{2}) / 5$ , a polynomial, and $ψ : = σ \circ q$ . Then $ψ$ is smooth by [F2], $0 \leq ψ \leq 1$ by [F1], and $ψ$ is radial because $q$ depends on $∣ ξ ∣$ only. Moreover $q (ξ) \geq 1 ⟺ ∣ ξ ∣^{2} \leq 1 ⟺ ∣ ξ ∣ \leq 1$ and $q (ξ) \leq 0 ⟺ ∣ ξ ∣^{2} \geq 9 / 4 ⟺ ∣ ξ ∣ \geq 3 / 2$ , so by [F1] $ψ (ξ) = 1$ for $∣ ξ ∣ \leq 1$ and $ψ (ξ) = 0$ for $∣ ξ ∣ \geq 3 / 2$ . Hence $supp ψ \subset {∣ ξ ∣ \leq 3 / 2} \subset {∣ ξ ∣ < 2}$ , $ψ$ is compactly supported, and $ψ \in C_{c}^{\infty} (R^{n}) \subset S$ by [F3]. This proves the existence clause and, since every subsequent step uses only the listed properties ( $0 \leq ψ \leq 1$ , $ψ = 1$ on $∣ ξ ∣ \leq 1$ , $ψ = 0$ for $∣ ξ ∣ \geq 3 / 2$ after the construction, or more generally $ψ$ vanishing for $∣ ξ ∣ \geq 2$ when only $supp ψ \subset {∣ ξ ∣ < 2}$ is assumed), the corresponding clauses for an arbitrary such $ψ$ .

2.1F1step 1.1algebra

The partition identity. Fix $ψ$ as in the statement and define $φ_{0} : = ψ$ , $φ_{j} : = ψ (2^{- j} ξ) - ψ (2^{- (j - 1)} ξ)$ for $j \geq 1$ ; each $φ_{j}$ is radial and lies in $C_{c}^{\infty} (R^{n})$ as a difference of rescalings of $ψ$ . Telescoping gives, for every $N \geq 0$ and every $ξ$ , $\sum_{j = 0}^{N} φ_{j} (ξ) = ψ (2^{- N} ξ),$ and $ψ (2^{- N} ξ) \to ψ (0) = 1$ as $N \to \infty$ because $ψ$ is continuous and $ψ = 1$ on the unit ball. Hence $\sum_{j \geq 0} φ_{j} (ξ) = 1$ for every $ξ$ . The sum is locally finite: if $∣ ξ ∣ \leq R$ and $j \geq 1$ with $2^{- (j - 1)} R \leq 1$ , then $∣ 2^{- j} ξ ∣ \leq ∣ 2^{- (j - 1)} ξ ∣ \leq 1$ and both $ψ$ values equal $1$ , so $φ_{j} (ξ) = 0$ ; thus only finitely many $j$ with $2^{j - 1} < R$ contribute on the ball of radius $R$ .

3.1step 1.1step 2.1F3algebra

Supports. For $j \geq 1$ put $u : = 2^{- j} ∣ ξ ∣$ , so that the two arguments have moduli $u$ and $2 u$ . If $2 u \leq 1$ then both moduli are at most $1$ and both $ψ$ values equal $1$ , so $φ_{j} (ξ) = 0$ ; if $u \geq 2$ then both moduli are at least $2$ and, since $supp ψ \subset {∣ ξ ∣ < 2}$ , both $ψ$ values vanish, so $φ_{j} (ξ) = 0$ . Therefore $φ_{j} (ξ) \neq 0$ forces $2^{j - 1} < ∣ ξ ∣ < 2^{j + 1}$ , which proves $supp φ_{j} \subset {2^{j - 1} \leq ∣ ξ ∣ \leq 2^{j + 1}}$ . The case $j = 0$ is $supp φ_{0} = supp ψ \subset {∣ ξ ∣ < 2} \subset {∣ ξ ∣ \leq 2}$ .

4.1step 2.1step 3.1algebra

Sign and overlap. We claim $φ_{j} \geq 0$ for every $j$ and every $ξ$ without any monotonicity hypothesis. For $j = 0$ this is $ψ \geq 0$ . For $j \geq 1$ keep $u = 2^{- j} ∣ ξ ∣$ : if $u \leq 1 / 2$ then both moduli are at most $1$ and $φ_{j} (ξ) = 0$ ; if $1 / 2 < u \leq 1$ then the smaller modulus $u$ gives $ψ (2^{- j} ξ) = 1$ and $φ_{j} (ξ) = 1 - ψ (2^{- (j - 1)} ξ) \in [0, 1]$ ; if $1 < u < 2$ then the larger modulus $2 u$ exceeds $2$ , so $ψ (2^{- (j - 1)} ξ) = 0$ and $φ_{j} (ξ) = ψ (2^{- j} ξ) \in [0, 1]$ ; finally if $u \geq 2$ then both moduli are at least $2$ and $φ_{j} (ξ) = 0$ . Thus every nonzero value lies in $[0, 1]$ , so $φ_{j} \geq 0$ and $φ_{j}^{2} \leq φ_{j}$ . Since $\sum_{j} φ_{j} = 1$ by step 2.1, summing the pointwise inequality gives $\sum_{j} φ_{j} (ξ)^{2} \leq 1$ . For the lower bound, at most three $φ_{j}$ are nonzero at any fixed $ξ$ : by the four cases above, a nonzero value requires $u = 2^{- j} ∣ ξ ∣ \in (1 / 2, 2)$ , and three consecutive halvings $u, u / 2, u / 4$ span the factor $4$ while the interval $(1 / 2, 2)$ has multiplicative length exactly $4$ , so at most two of the values $2^{- j} ∣ ξ ∣$ lie in $(1 / 2, 2)$ (in particular at most three). Hence, by Cauchy-Schwarz at the fixed $ξ$ , $1 = (\sum_{j} φ_{j} (ξ))^{2} \leq (\sum_{j} 1_{φ_{j} (ξ) \neq 0}) (\sum_{j} φ_{j} (ξ)^{2}) \leq 3 \sum_{j} φ_{j} (ξ)^{2},$ which gives $\sum_{j} φ_{j} (ξ)^{2} \geq 1 / 3$ .

4.2F3F4step 3.1algebra

Derivative bounds. Fix a multi-index $α$ and put $C_{α} : = (1 + 2^{∣ α ∣}) ∥ \partial^{α} ψ ∥_{\infty} < \infty$ ; the value is finite because $ψ \in C_{c}^{\infty}$ has bounded derivatives. For $j \geq 1$ , [F4] with $R = 2^{j}$ and $h = ψ$ gives $\partial^{α} (ψ (2^{- j} ξ)) = 2^{- j ∣ α ∣} (\partial^{α} ψ) (2^{- j} ξ)$ , and similarly with $R = 2^{j - 1}$ ; hence $∣ \partial^{α} φ_{j} (ξ) ∣ \leq 2^{- j ∣ α ∣} ∥ \partial^{α} ψ ∥_{\infty} + 2^{- (j - 1) ∣ α ∣} ∥ \partial^{α} ψ ∥_{\infty} = C_{α} 2^{- j ∣ α ∣},$ while $∣ \partial^{α} φ_{0} (ξ) ∣ = ∣ \partial^{α} ψ (ξ) ∣ \leq C_{α}$ . On the support of $φ_{j}$ ( $j \geq 1$ ) step 3.1 gives $∣ ξ ∣ \leq 2^{j + 1}$ , so $2^{- j ∣ α ∣} \leq 2^{∣ α ∣} ∣ ξ ∣^{- ∣ α ∣}$ and therefore $∣ \partial^{α} φ_{j} (ξ) ∣ \leq 2^{∣ α ∣} C_{α} ∣ ξ ∣^{- ∣ α ∣}$ for $ξ \neq 0$ ; off the support the left-hand side is zero.

5.1step 1.1step 2.1step 3.1step 4.1step 4.2∎

The four numbered clauses are steps 2.1 (partition), 3.1 (supports), 4.1 (sign, overlap, square sums) and 4.2 (derivative bounds and their consequence), and the existence clause with the strict support is step 1.1. Since steps 2.1 to 4.2 used only the properties $ψ \in C_{c}^{\infty}$ , $0 \leq ψ \leq 1$ , $ψ = 1$ on $∣ ξ ∣ \leq 1$ and $supp ψ \subset {∣ ξ ∣ < 2}$ (the last only through $ψ = 0$ for $∣ ξ ∣ \geq 2$ ), the conclusions hold for every $ψ$ with those properties.

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators

Definition

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Fix a function $ψ$ and its partition $(φ_{j})_{j \geq 0}$ as in Existence of a smooth inhomogeneous dyadic frequency partition. Define the companion sequence by ${\tilde{φ}}_{j} : = φ_{j - 1} + φ_{j} + φ_{j + 1} (j \geq 0),$ with $φ_{- 1} : = 0$ . For $f \in S^{'} (R^{n})$ define $Δ_{j} f : = F^{- 1} (φ_{j} \cdot F f), {\tilde{Δ}}_{j} f : = F^{- 1} ({\tilde{φ}}_{j} \cdot F f),$ where $φ_{j} \cdot F f$ and ${\tilde{φ}}_{j} \cdot F f$ are the products of the tempered distribution $F f$ with the smooth polynomially bounded symbols $φ_{j}, {\tilde{φ}}_{j}$ (transposition, Smooth polynomially bounded multipliers on schwartz space); for $f \in S$ these operators are the translation-invariant Fourier multipliers $T_{φ_{j}} f$ , $T_{{\tilde{φ}}_{j}} f$ of Translation-invariant Fourier multiplier on the Schwartz core; for $f \in L^{p} (R^{n}; C)$ , $1 \leq p < \infty$ , define $Δ_{j} f : = f * K_{j}$ and ${\tilde{Δ}}_{j} f : = f * {\tilde{K}}_{j}$ , where $K_{j} : = F^{- 1} φ_{j}, {\tilde{K}}_{j} : = F^{- 1} {\tilde{φ}}_{j}$ are the inverse transforms of the symbols (Fourier transform of a tempered distribution). The following well-definedness and compatibility facts are part of the definition and are recorded with their cited suppliers.

Domains. Each $φ_{j}$ and each ${\tilde{φ}}_{j}$ is smooth, compactly supported and bounded together with all its derivatives, hence a smooth polynomially bounded multiplier: for $f \in S (R^{n})$ the products $φ_{j} \hat{f}$ and ${\tilde{φ}}_{j} \hat{f}$ lie in $S (R^{n})$ (Smooth polynomially bounded multipliers on schwartz space), so $S \subset D_{φ_{j}} \cap D_{{\tilde{φ}}_{j}}$ and $T_{φ_{j}} f$ , $T_{{\tilde{φ}}_{j}} f$ are well-defined tempered distributions. Because $φ_{0} = ψ$ and $φ_{j} (ξ) = ψ (2^{- j} ξ) - ψ (2^{- (j - 1)} ξ)$ for $j \geq 1$ , the pieces satisfy the rescaling law $φ_{k + j - 1} (ξ) = φ_{k} (2^{- (j - 1)} ξ)$ for every $k \geq 1$ and $j \geq 1$ (so $φ_{j} (ξ) = φ_{1} (2^{- (j - 1)} ξ)$ for $j \geq 1$ ); the companion symbols are sums of neighbouring pieces, and no single rescaling law for all $j \geq 0$ is used. They have the recorded supports $supp {\tilde{φ}}_{j} \subset {∣ ξ ∣ \leq 2^{j + 2}}$ for every $j \geq 0$ , with $supp {\tilde{φ}}_{j} \subset {2^{j - 2} \leq ∣ ξ ∣ \leq 2^{j + 2}}$ for $j \geq 2$ , together with $supp {\tilde{φ}}_{0} \subset {∣ ξ ∣ \leq 4}$ . The vanishing is strict: for $j \geq 1$ , since $ψ = 1$ on the unit ball, $φ_{j} (ξ) = 0$ whenever $∣ ξ ∣ \leq 2^{j - 1}$ (both arguments of $ψ$ have modulus at most $1$ ), and since $ψ = 0$ for $∣ ξ ∣ \geq 2$ , $φ_{j} (ξ) = 0$ whenever $∣ ξ ∣ \geq 2^{j + 1}$ ; thus for $j \geq 1$ its nonzero set is contained in the open annulus $2^{j - 1} < ∣ ξ ∣ < 2^{j + 1}$ , while its closed support lies in $2^{j - 1} \leq ∣ ξ ∣ \leq 2^{j + 1}$ . Boundary points can belong to the support even though the function vanishes there, and $φ_{0} = 0$ for $∣ ξ ∣ \geq 2$ .
The $L^{p}$ definition. $K_{j}, {\tilde{K}}_{j} \in S (R^{n})$ (Fourier transform acts continuously on Schwartz space), hence $K_{j}, {\tilde{K}}_{j} \in L^{1} (R^{n})$ , and Young's inequality (Young's convolution inequality under Countable Choice) shows that for $f \in L^{p}$ , $1 \leq p < \infty$ , the convolutions $f * K_{j}$ , $f * {\tilde{K}}_{j}$ are defined almost everywhere, lie in $L^{p}$ and satisfy $∥ f * K_{j} ∥_{p} \leq ∥ K_{j} ∥_{1} ∥ f ∥_{p}$ , $∥ f * {\tilde{K}}_{j} ∥_{p} \leq ∥ {\tilde{K}}_{j} ∥_{1} ∥ f ∥_{p}$ ; the convolution is the one of Convolution of two functions on $R^{n}$ .
Agreement on $S$ and composition. For $f \in S$ the convolution $K_{j} * f$ is Schwartz by Schwartz convolution and product laws, hence lies in $L^{1}$ , its integral transform is $\hat{K_{j} * f} = {\hat{K}}_{j} \hat{f} = φ_{j} \hat{f}$ by the convolution theorem (Fourier transform turns L1 convolution into multiplication), and Fourier inversion (Fourier inversion on Schwartz space) gives $K_{j} * f = F^{- 1} (φ_{j} \hat{f})$ as functions; since $T_{φ_{j}} f = F^{- 1} (u_{φ_{j} \hat{f}})$ and the right side is the regular distribution of $F^{- 1} (φ_{j} \hat{f})$ , the two definitions of $Δ_{j} f$ agree on $S$ , and $\hat{Δ_{j} f} = φ_{j} \hat{f}$ as tempered distributions (using that the distributional transform agrees with the integral transform on $L^{1}$ functions, Fourier transform agrees with l one and plancherel transforms); the same holds for the companions. Moreover, if $m, n$ are smooth polynomially bounded symbols with $S \subset D_{m} \cap D_{n}$ , then $T_{m} T_{n} = T_{m n}$ on $S$ : for $f \in S$ the density $n \hat{f}$ lies in $S$ , so $T_{n} f$ is the regular distribution of $F^{- 1} (n \hat{f}) \in S$ and $T_{m} (T_{n} f) = T_{m n} f$ by the same computation.
The companion identity and the low-frequency block. Because $φ_{j} φ_{k} = 0$ pointwise whenever $∣ j - k ∣ \geq 2$ (the annular supports meet at most at the endpoint spheres, where both factors vanish), expanding $1 = (\sum_{j} φ_{j})^{2}$ gives $1 = \sum_{j} φ_{j}^{2} + 2 \sum_{j} φ_{j} φ_{j + 1} = \sum_{j} (φ_{j - 1} + φ_{j} + φ_{j + 1}) φ_{j} = \sum_{j} {\tilde{φ}}_{j} φ_{j}$ , the sums being locally finite. Neither the low-frequency block $Δ_{0}$ nor its companion ${\tilde{Δ}}_{0}$ is assigned mean zero: indeed $\int_{R^{n}} K_{j} = φ_{j} (0)$ and $\int_{R^{n}} {\tilde{K}}_{j} = {\tilde{φ}}_{j} (0)$ , so $\int K_{0} = ψ (0) = 1$ and $\int {\tilde{K}}_{0} = φ_{0} (0) + φ_{1} (0) = 1$ , and no cancellation is claimed for these blocks.

This partition is fixed once and for all on this page. The operator norms of $Δ_{j}$ on $L^{2}$ are at most $∥ φ_{j} ∥_{\infty} \leq 1$ (Exact L2 Fourier multiplier norm), and all constants below refer to this fixed partition.

LemmaStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Dyadic pieces have annular Fourier support and uniformly bounded rescaled kernels

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). With the fixed partition and notation of The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators:

for $j \geq 1$ the kernel $K_{j} = F^{- 1} φ_{j}$ lies in $S (R^{n})$ and satisfies $K_{j} (x) = 2^{(j - 1) n} K_{1} (2^{j - 1} x) (x \in R^{n})$ and $\int_{R^{n}} K_{j} = 0$ ; also ${\tilde{K}}_{j} \in S$ , with ${\tilde{K}}_{j} = K_{j - 1} + K_{j} + K_{j + 1}$ (a finite sum of Schwartz kernels; no scaling law for the companions);
for every $N \geq 0$ there are constants $C_{N}, C_{N}^{'} < \infty$ depending only on $n, ψ, N$ with $∣ K_{j} (x) ∣ \leq C_{N} 2^{j n} (1 + 2^{j} ∣ x ∣)^{- N}, ∣ {\tilde{K}}_{j} (x) ∣ \leq C_{N}^{'} 2^{j n} (1 + 2^{j} ∣ x ∣)^{- N}$ for all $j \geq 1$ and $x \in R^{n}$ , while $K_{0}, {\tilde{K}}_{0} \in S$ ;
consequently $∥ K_{j} ∥_{1} \leq C$ and $∥ {\tilde{K}}_{j} ∥_{1} \leq C$ uniformly in $j$ , and for $f \in L^{p} (R^{n}; C)$ , $1 \leq p \leq \infty$ , $∥ f * K_{j} ∥_{p} \leq C ∥ f ∥_{p}, ∥ f * {\tilde{K}}_{j} ∥_{p} \leq C ∥ f ∥_{p}$ uniformly in $j \geq 0$ , that is $∥ Δ_{j} f ∥_{p} \leq C ∥ f ∥_{p}$ and $∥ {\tilde{Δ}}_{j} f ∥_{p} \leq C ∥ f ∥_{p}$ .

Facts & Assumptions

Given: the fixed partition $(φ_{j})$ of Existence of a smooth inhomogeneous dyadic frequency partition with its companion sequence $({\tilde{φ}}_{j})$ and the kernels $K_{j} = F^{- 1} φ_{j}$ , ${\tilde{K}}_{j} = F^{- 1} {\tilde{φ}}_{j}$ of The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators; an integer $N \geq 0$ .

[F1]

The symbols satisfy $φ_{0} = ψ$ , $φ_{j} (ξ) = ψ (2^{- j} ξ) - ψ (2^{- (j - 1)} ξ)$ for $j \geq 1$ and ${\tilde{φ}}_{j} = φ_{j - 1} + φ_{j} + φ_{j + 1}$ with $φ_{- 1} = 0$ ; $ψ$ is Schwartz, $ψ (0) = 1$ and $ψ = 0$ for $∣ ξ ∣ \geq 2$ ; and $φ_{j} (ξ) = φ_{1} (2^{- (j - 1)} ξ)$ for $j \geq 1$ (Existence of a smooth inhomogeneous dyadic frequency partition, The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators).

[F2]

$K_{j}, {\tilde{K}}_{j} \in S (R^{n})$ , $F K_{j} = φ_{j}$ and $F {\tilde{K}}_{j} = {\tilde{φ}}_{j}$ (Fourier transform acts continuously on Schwartz space); for $h \in S (R^{n})$ the inversion formula $h (x) = \int_{R^{n}} \hat{h} (ξ) e^{2 π i x \cdot ξ} d ξ$ holds with an absolutely convergent integral (Fourier inversion on Schwartz space), and for $L^{1}$ functions the distributional transform is the regular distribution of the integral transform (Fourier transform agrees with l one and plancherel transforms), so the integral transform of the integrable function $K_{j}$ equals $φ_{j}$ pointwise.

[F3]

For $n \geq 1$ , if $h \in S (R^{n})$ then $h \in L^{1} (R^{n})$ (Schwartz derivatives are integrable), and for every $N \geq 0$ the quantity $\sup_{u \in R^{n}} (1 + ∣ u ∣)^{N} ∣ h (u) ∣$ is finite and bounded by a finite sum of the seminorms $p_{α 0} (h)$ , because $(1 + ∣ u ∣)^{N} \leq C_{N} \sum_{∣ α ∣ \leq N} ∣ u^{α} ∣$ (Schwartz space and its seminorms, $C^{k}$ maps and multi-index derivative notation in Euclidean space).

[F4]

Young's inequality: for $1 \leq p \leq \infty$ , $f \in L^{p}$ and $g \in L^{1}$ the convolution $f * g$ is defined almost everywhere and $∥ f * g ∥_{p} \leq ∥ f ∥_{p} ∥ g ∥_{1}$ (Young's convolution inequality under Countable Choice).

Proof

technique · direct

1.1F1algebra

Rescaling law for the pieces. For $j \geq 1$ and every $ξ$ , $φ_{j} (ξ) = ψ (2^{- j} ξ) - ψ (2^{- (j - 1)} ξ) = φ_{1} (2^{- (j - 1)} ξ)$ , because $φ_{1} (2^{- (j - 1)} ξ) = ψ (2^{- 1} 2^{- (j - 1)} ξ) - ψ (2^{- (j - 1)} ξ)$ , and more generally $φ_{k + j - 1} (ξ) = φ_{k} (2^{- (j - 1)} ξ)$ for every $k \geq 1$ by the same computation with $k$ in place of $1$ . The companions are sums, not rescalings: ${\tilde{φ}}_{j} = φ_{j - 1} + φ_{j} + φ_{j + 1}$ , hence ${\tilde{K}}_{j} = K_{j - 1} + K_{j} + K_{j + 1}$ by linearity of the inverse transform, and no rescaling law is claimed for them.

2.1F1F2step 1.1algebra

Rescaling law for the kernels. Since $K_{1} = F^{- 1} φ_{1} \in S$ and $K_{j} = F^{- 1} φ_{j} = F^{- 1} (φ_{1} (2^{- (j - 1)} \cdot))$ by step 1.1, the inversion formula applied at $x$ gives $K_{j} (x) = \int φ_{1} (2^{- (j - 1)} ξ) e^{2 π i x \cdot ξ} d ξ$ ; substituting $ξ = 2^{j - 1} η$ , $d ξ = 2^{(j - 1) n} d η$ , this becomes $K_{j} (x) = 2^{(j - 1) n} \int φ_{1} (η) e^{2 π i (2^{j - 1} x) \cdot η} d η = 2^{(j - 1) n} K_{1} (2^{j - 1} x)$ . For the companions, step 1.1 gives ${\tilde{K}}_{j} = K_{j - 1} + K_{j} + K_{j + 1}$ (the inverse transform is linear and each $K_{i}$ lies in $S$ ); a finite sum of Schwartz kernels again lies in $S$ .

3.1F1F2step 2.1algebra

Mean zero of the high-frequency kernels. For $j \geq 1$ the integral ${\hat{K}}_{j} (0) = \int_{R^{n}} K_{j} (x) d x$ is the value at the origin of the integral transform of $K_{j}$ , which by [F2] equals $φ_{j} (0)$ ; since $ψ (0) = 1$ by [F1], $φ_{j} (0) = ψ (0) - ψ (0) = 0$ .

3.2F2F3step 2.1algebra

Pointwise decay. Fix $N \geq 0$ . By [F3] applied to $K_{1}$ there is $A_{N} < \infty$ , depending only on $n, ψ, N$ , with $∣ K_{1} (u) ∣ \leq A_{N} (1 + ∣ u ∣)^{- N}$ for all $u$ ; since $1 + 2^{j} ∣ x ∣ \leq 2 (1 + 2^{j - 1} ∣ x ∣)$ for $j \geq 1$ , step 2.1 gives, for $j \geq 1$ , $∣ K_{j} (x) ∣ = 2^{(j - 1) n} ∣ K_{1} (2^{j - 1} x) ∣ \leq A_{N} 2^{(j - 1) n} (1 + 2^{j - 1} ∣ x ∣)^{- N} \leq A_{N} 2^{N + n} 2^{j n} (1 + 2^{j} ∣ x ∣)^{- N}$ , so the asserted bound holds with $C_{N} : = A_{N} 2^{N + n}$ . For the companions, step 2.1 writes ${\tilde{K}}_{j} = K_{j - 1} + K_{j} + K_{j + 1}$ ; summing the bounds just proved for these three kernels, and for $j = 1$ also the Schwartz bound of $K_{0}$ from [F3], gives $∣ {\tilde{K}}_{j} (x) ∣ \leq C_{N}^{'} 2^{j n} (1 + 2^{j} ∣ x ∣)^{- N}$ after absorbing the fixed factors $3$ , $2^{N}$ and $2^{n}$ into $C_{N}^{'}$ . Since $K_{0} = F^{- 1} ψ$ and ${\tilde{K}}_{0} = F^{- 1} {\tilde{φ}}_{0}$ are inverse transforms of Schwartz functions, both lie in $S$ by [F2].

4.1F3step 3.2algebra

Uniform $L^{1}$ bounds. Take $N = n + 1$ in step 3.2 and substitute $u = 2^{j} x$ : for $j \geq 1$ , $\int ∣ K_{j} ∣ \leq C_{N} 2^{j n} \int (1 + 2^{j} ∣ x ∣)^{- N} d x = C_{N} \int (1 + ∣ u ∣)^{- N} d u = : C < \infty$ , and likewise $\int ∣ {\tilde{K}}_{j} ∣ \leq C_{N}^{'} \int (1 + ∣ u ∣)^{- N} d u \leq C$ after enlarging $C$ ; the integrals are finite: on $∣ u ∣ \leq 1$ the integrand is bounded and the ball lies in a finite-volume box; on $2^{k} < ∣ u ∣ \leq 2^{k + 1}$ its integral is at most $2^{- k (n + 1)} (2^{k + 2})^{n} = 2^{2 n - k}$ , and $\sum_{k \geq 0} 2^{- k} < \infty$ (A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included). For $j = 0$ , [F3] gives $∥ K_{0} ∥_{1} + ∥ {\tilde{K}}_{0} ∥_{1} < \infty$ , and these two constants are absorbed into $C$ .

5.1F4step 4.1algebra

Uniform $L^{p}$ bounds. For $f \in L^{p}$ , $1 \leq p \leq \infty$ , Young's inequality [F4] applied to $f$ and $K_{j} \in L^{1}$ gives $∥ f * K_{j} ∥_{p} \leq ∥ K_{j} ∥_{1} ∥ f ∥_{p} \leq C ∥ f ∥_{p}$ , and likewise for ${\tilde{K}}_{j}$ , uniformly in $j \geq 0$ .

6.1step 1.1step 2.1step 3.1step 3.2step 4.1step 5.1∎

Clauses 1, 2 and 3 are steps 2.1 with 3.1, step 3.2 and steps 4.1 with 5.1, respectively.

LemmaStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Dyadic pieces are uniformly Mihlin multipliers and uniformly Lp-bounded

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). For every $j \geq 0$ the symbol $φ_{j}$ of The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators is a Mihlin symbol in the sense of Mihlin smoothness convention above half the dimension, with constants $C_{α} \leq A_{α} (n, ψ)$ for $∣ α ∣ \leq ⌊ n / 2 ⌋ + 1$ that do not depend on $j$ ; the same holds for the companion symbols ${\tilde{φ}}_{j}$ . Consequently, for every $1 < p < \infty$ the multiplier operators $Δ_{j}$ and ${\tilde{Δ}}_{j}$ extend uniquely to bounded operators on $L^{p} (R^{n}; C)$ with $∥ Δ_{j} f ∥_{p} \leq C_{n, ψ} \max (p, (p - 1)^{- 1}) ∥ f ∥_{p}, ∥ {\tilde{Δ}}_{j} f ∥_{p} \leq C_{n, ψ}^{'} \max (p, (p - 1)^{- 1}) ∥ f ∥_{p}$ for all $f \in L^{p}$ and all $j \geq 0$ ; on $S$ the extensions agree with the convolution representatives $f * K_{j}$ , $f * {\tilde{K}}_{j}$ . In particular each $Δ_{j}$ is well defined on $L^{p}$ as an honest function given by that convolution.

Facts & Assumptions

Given: the fixed partition $(φ_{j})$ with companions $({\tilde{φ}}_{j})$ and kernels $K_{j}, {\tilde{K}}_{j}$ of The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators; the exponent $q : = ⌊ n / 2 ⌋ + 1$ of Mihlin smoothness convention above half the dimension; a real $1 < p < \infty$ .

[F1]

$φ_{j}, {\tilde{φ}}_{j} \in C_{c}^{\infty} (R^{n})$ and, for every multi-index $α$ , $∣ \partial^{α} φ_{j} (ξ) ∣ \leq C_{α} 2^{- j ∣ α ∣}$ for $j \geq 1$ and all $ξ$ , with $∣ \partial^{α} φ_{0} (ξ) ∣ \leq C_{α}$ ; moreover $∣ \partial^{α} φ_{j} (ξ) ∣ \leq 2^{∣ α ∣} C_{α} ∣ ξ ∣^{- ∣ α ∣}$ for $j \geq 1$ and $ξ \neq 0$ (Existence of a smooth inhomogeneous dyadic frequency partition).

[F2]

$supp ψ \subset {∣ ξ ∣ < 2}$ and $supp φ_{j} \subset {2^{j - 1} \leq ∣ ξ ∣ \leq 2^{j + 1}}$ for $j \geq 1$ , $supp φ_{0} \subset {∣ ξ ∣ \leq 2}$ ; hence all derivatives of $φ_{0}$ vanish for $∣ ξ ∣ \geq 2$ (Existence of a smooth inhomogeneous dyadic frequency partition).

[F3]

Mihlin's theorem: if $m$ is a Mihlin symbol with constants $C_{α}$ , $A : = \max_{∣ α ∣ \leq q} C_{α}$ , then $m$ is an $L^{p}$ Fourier multiplier for $1 < p < \infty$ and $∥ m ∥_{M_{p}} \leq C_{n} \max (p, (p - 1)^{- 1}) (A + ∥ m ∥_{\infty})$ (The Mihlin–Hörmander Fourier multiplier theorem, Lp Fourier multiplier and its norm, Translation-invariant Fourier multiplier on the Schwartz core).

[F4]

For $f \in S$ the tempered distribution $T_{φ_{j}} f$ is the regular distribution of the convolution $f * K_{j}$ , and $K_{j} \in L^{1}$ with $∥ K_{j} ∥_{1} \leq C$ uniformly; hence $f \mapsto f * K_{j}$ is a bounded operator on $L^{p}$ with norm at most $C$ , and the compactly supported smooth functions are dense in $L^{p}$ for finite $p$ , so bounded operators agreeing on $S$ agree everywhere by uniqueness of the bounded extension (The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators, Dyadic pieces have annular Fourier support and uniformly bounded rescaled kernels, Young's convolution inequality under Countable Choice, Complex finite-simple and smooth compact-support density for finite p, A bounded linear map from a dense normed subspace into a Banach space extends uniquely with the same norm).

Proof

technique · direct

1.1F1F2algebra

Uniform Mihlin constants. Fix a multi-index $α$ with $∣ α ∣ \leq q$ . For $j \geq 1$ and $ξ \neq 0$ , [F1] and [F2] give $∣ \partial^{α} φ_{j} (ξ) ∣ \leq 2^{∣ α ∣} C_{α} ∣ ξ ∣^{- ∣ α ∣}$ , and for $j = 0$ the same bound holds with constant $2^{∣ α ∣} C_{α}$ : for $∣ ξ ∣ < 2$ one has $2^{∣ α ∣} ∣ ξ ∣^{- ∣ α ∣} \geq 1$ so the bound follows from $∣ \partial^{α} φ_{0} ∣ \leq C_{α}$ , while for $∣ ξ ∣ \geq 2$ all derivatives of $φ_{0}$ vanish by [F2]. Hence $φ_{j}$ is a Mihlin symbol with constants $A_{α} : = 2^{∣ α ∣} C_{α}$ for every $j \geq 0$ . For the companions, $∣ \partial^{α} {\tilde{φ}}_{j} ∣ \leq ∣ \partial^{α} φ_{j - 1} ∣ + ∣ \partial^{α} φ_{j} ∣ + ∣ \partial^{α} φ_{j + 1} ∣ \leq 3 A_{α} ∣ ξ ∣^{- ∣ α ∣}$ for $ξ \neq 0$ (each summand satisfying the same bound, with $φ_{- 1} = 0$ for $j = 0$ ), so the constants $A_{α}^{'} : = 3 A_{α}$ are uniform in $j$ as well.

2.1F3step 1.1algebra

Uniform $L^{p}$ multiplier bounds. By step 1.1 the symbols $φ_{j}, {\tilde{φ}}_{j}$ are Mihlin symbols with constants bounded by the $j$ -independent numbers $A : = \max_{∣ α ∣ \leq q} A_{α}$ and $A^{'} = \max_{∣ α ∣ \leq q} A_{α}^{'}$ , and $∣ m ∣ \leq ∥ m ∥_{\infty} \leq 1$ for both. [F3] therefore makes each of them an $L^{p}$ Fourier multiplier with $∥ φ_{j} ∥_{M_{p}} \leq C_{n} \max (p, (p - 1)^{- 1}) (A + 1)$ and $∥ {\tilde{φ}}_{j} ∥_{M_{p}} \leq C_{n} \max (p, (p - 1)^{- 1}) (A^{'} + 1)$ , uniformly in $j$ , and the corresponding operators $T_{φ_{j}}, T_{{\tilde{φ}}_{j}}$ act boundedly on $L^{p}$ by the definition of the multiplier norm.

3.1F4step 2.1algebra

The extension is the convolution. Fix $j \geq 0$ . On $S$ the operator $T_{φ_{j}}$ agrees with the convolution representative $f \mapsto f * K_{j}$ by [F4], and the convolution operator is bounded on $L^{p}$ with norm at most $C$ by Young's inequality [F4]; since $S$ is dense in $L^{p}$ ( $p$ finite) and the bounded extension of $T_{φ_{j}}$ is unique, the $L^{p}$ multiplier operator equals the convolution operator, so for every $f \in L^{p}$ the class $Δ_{j} f$ has the honest representative $f * K_{j}$ and $∥ Δ_{j} f ∥_{p} \leq C ∥ f ∥_{p}$ ; the same argument applies to ${\tilde{K}}_{j}$ .

4.1step 1.1step 2.1step 3.1∎

Conclusion. Steps 1.1 and 2.1 give the uniform Mihlin property and the uniform multiplier bounds, and step 3.1 identifies the extensions with the convolution representatives; the asserted inequalities follow with $C_{n, ψ}$ a constant depending only on $n, ψ$ (absorbing $C_{n} (A + 1)$ and the $L^{1}$ bound, and enlarging it for the companions).

LemmaStatement: Literature-sourcedProof: AI-adaptedOpen item page →

L2 almost orthogonality of the dyadic pieces

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). For every $f \in L^{2} (R^{n}; C)$ , $\sum_{j \geq 0} ∥ Δ_{j} f ∥_{L^{2}}^{2} \leq ∥ f ∥_{L^{2}}^{2} \leq 3 \sum_{j \geq 0} ∥ Δ_{j} f ∥_{L^{2}}^{2} .$ In particular $\sum_{j \geq 0} ∥ Δ_{j} f ∥_{L^{2}}^{2} < \infty$ . The constants $1$ and $3$ depend on no parameter beyond the fixed partition.

Facts & Assumptions

Given: the fixed partition $(φ_{j})$ and operators $Δ_{j}$ of The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators; a function $f \in L^{2} (R^{n}; C)$ ; the Plancherel isometry $F_{2}$ and the complex $L^{2}$ conventions of Complex Lp classes and Euclidean test-function conventions.

[F1]

Each $φ_{j} \in C_{c}^{\infty} (R^{n})$ satisfies $0 \leq φ_{j} \leq 1$ , the pointwise bounds $\frac{1}{3} \leq \sum_{j \geq 0} φ_{j} (ξ)^{2} \leq 1$ with a locally finite sum, and for $f \in S$ the function $Δ_{j} f$ lies in $S$ with $\hat{Δ_{j} f} = φ_{j} \hat{f}$ (Existence of a smooth inhomogeneous dyadic frequency partition, The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators).

[F2]

Plancherel: $F_{2} : L^{2} (R^{n}; C) \to L^{2} (R^{n}; C)$ is a surjective isometry preserving the first-variable-linear inner product, so $∥ g ∥_{2}^{2} = \int ∣ F_{2} g ∣^{2}$ and $⟨ g, h ⟩ = ⟨ F_{2} g, F_{2} h ⟩$ (Plancherel theorem).

[F3]

The multiplier by a symbol $m$ with $∥ m ∥_{\infty} \leq M$ that belongs to $C_{c}^{\infty}$ extends uniquely from $S$ to a bounded operator on $L^{2}$ of norm at most $M$ , given by $F_{2}^{- 1} (m F_{2} g)$ (Exact L2 Fourier multiplier norm).

[F4]

For $g \in L^{2}$ and $j \geq 0$ , the convolution representative $Δ_{j} g = g * K_{j}$ lies in $L^{2}$ with $∥ Δ_{j} g ∥_{2} \leq ∥ K_{j} ∥_{1} ∥ g ∥_{2} \leq C ∥ g ∥_{2}$ , the constant being uniform in $j$ (Young's convolution inequality under Countable Choice, Dyadic pieces have annular Fourier support and uniformly bounded rescaled kernels).

[F5]

The smooth compactly supported functions are dense in $L^{2} (R^{n}; C)$ (Complex finite-simple and smooth compact-support density for finite p): there is a sequence $f_{k} \in C_{c}^{\infty} \subset S$ with $f_{k} \to f$ in $L^{2}$ .

[F6]

Tonelli's theorem for nonnegative measurable functions on sigma-finite products, in particular for summation in a discrete index against Lebesgue measure (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product).

Proof

technique · direct

1.1F1F2algebra

The Schwartz case. Let $f \in S$ . For every $j$ , [F1] gives $Δ_{j} f \in S$ with $\hat{Δ_{j} f} = φ_{j} \hat{f}$ ; by Plancherel [F2] applied to $g = Δ_{j} f$ and to $f$ , $∥ Δ_{j} f ∥_{2}^{2} = \int_{R^{n}} ∣ \hat{Δ_{j} f} ∣^{2} = \int_{R^{n}} φ_{j}^{2} ∣ \hat{f} ∣^{2}$ and $∥ f ∥_{2}^{2} = \int ∣ \hat{f} ∣^{2}$ .

2.1F1F2F6step 1.1algebra

Summing the Schwartz identity. With $f \in S$ as in step 1.1, the series $\sum_{j} φ_{j} (ξ)^{2}$ is locally finite with $\sum_{j} φ_{j}^{2} \in [1 / 3, 1]$ pointwise by [F1], so Tonelli's theorem [F6] applied to the nonnegative functions $∣ φ_{j} \hat{f} ∣^{2}$ gives $\sum_{j \geq 0} ∥ Δ_{j} f ∥_{2}^{2} = \int_{R^{n}} (\sum_{j \geq 0} φ_{j} (ξ)^{2}) ∣ \hat{f} (ξ) ∣^{2} d ξ$ , and the pointwise bounds sandwich this between $\frac{1}{3} \int ∣ \hat{f} ∣^{2} = \frac{1}{3} ∥ f ∥_{2}^{2}$ and $\int ∣ \hat{f} ∣^{2} = ∥ f ∥_{2}^{2}$ . This proves the two-sided estimate, and the finiteness of $\sum_{j} ∥ Δ_{j} f ∥_{2}^{2}$ , for Schwartz $f$ .

2.2F3F4F5step 1.1algebra

The identity $\hat{Δ_{j} g} = φ_{j} \hat{g}$ passes to $L^{2}$ . For $j \geq 0$ both maps $g \mapsto Δ_{j} g = g * K_{j}$ and $g \mapsto F_{2}^{- 1} (φ_{j} F_{2} g)$ are bounded linear operators on $L^{2}$ by [F3] and [F4], and they agree on the dense subspace $S$ by step 1.1 and [F3]; given $g \in L^{2}$ and a sequence $g_{k} \in S$ with $g_{k} \to g$ from [F5], both operators applied to $g_{k}$ converge in $L^{2}$ to their values at $g$ , so $Δ_{j} g = F_{2}^{- 1} (φ_{j} F_{2} g)$ and hence $∥ Δ_{j} g ∥_{2}^{2} = \int φ_{j}^{2} ∣ F_{2} g ∣^{2}$ for every $j$ .

3.1F1F2F6step 2.2algebra∎

Conclusion. For $g \in L^{2}$ , step 2.2 gives $∥ Δ_{j} g ∥_{2}^{2} = \int φ_{j}^{2} ∣ F_{2} g ∣^{2}$ for every $j$ , and Tonelli [F6] then gives $\sum_{j \geq 0} ∥ Δ_{j} g ∥_{2}^{2} = \int (\sum_{j} φ_{j}^{2}) ∣ F_{2} g ∣^{2}$ ; the pointwise bounds $\frac{1}{3} \leq \sum_{j} φ_{j}^{2} \leq 1$ and $∥ g ∥_{2}^{2} = \int ∣ F_{2} g ∣^{2}$ from Plancherel [F2] yield $\frac{1}{3} ∥ g ∥_{2}^{2} \leq \sum_{j} ∥ Δ_{j} g ∥_{2}^{2} \leq ∥ g ∥_{2}^{2}$ , which is the stated two-sided estimate, and the finiteness of the sum.

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

The Littlewood-Paley square function

Definition

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). With the fixed partition and operators of The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators, for $f \in L^{p} (R^{n}; C)$ with $1 \leq p < \infty$ and $N \geq 0$ define, for $x \in R^{n}$ , $S_{N} f (x) : = (\sum_{j = 0}^{N - 1} ∣ Δ_{j} f (x) ∣^{2})^{1 / 2}, S f (x) : = (\sum_{j \geq 0} ∣ Δ_{j} f (x) ∣^{2})^{1 / 2} \in [0, \infty] .$ The functions $Δ_{j} f$ in these formulae are the convolution representatives $f * K_{j}$ of Dyadic pieces have annular Fourier support and uniformly bounded rescaled kernels. The following conventions and well-definedness facts are part of the definition.

Each $Δ_{j} f = f * K_{j}$ lies in $L^{p}$ by Dyadic pieces have annular Fourier support and uniformly bounded rescaled kernels. Its integral is defined at every $x$ : Holder gives $\int ∣ f (u) K_{j} (x - u) ∣ d u \leq ∥ f ∥_{p} ∥ K_{j} ∥_{p^{'}}$ , with $p^{'} = \infty$ when $p = 1$ (Complex Holder, Minkowski, and the quotient norm). Translations of a Schwartz kernel are continuous in $L^{p^{'}}$ : for finite $p^{'}$ use dominated convergence with a common Schwartz majorant, and for $p^{'} = \infty$ use its bounded first derivatives. Holder therefore makes this everywhere convolution representative continuous, hence Borel measurable (Dominated convergence, Borel measurable and Lebesgue measurable functions on $R^{n}$ ). Finite sums, products and the square root of nonnegative measurable functions preserve measurability (Arithmetic and lattice operations preserve measurability whenever they are defined), so every $S_{N} f$ is measurable and $S_{N} f \leq S_{N + 1} f$ pointwise.
$S f = \sup_{N} S_{N} f$ is measurable as the increasing limit of measurable functions, with values in $[0, \infty]$ ; no finiteness is asserted, that is $S f (x) = + \infty$ is allowed a priori.
Replacing $f$ by an almost everywhere equal function in $L^{p}$ replaces every convolution representative $Δ_{j} f$ by an almost everywhere equal function, hence replaces $S f$ by an almost everywhere equal function; the functional is therefore defined on almost everywhere classes, and $S f$ is recorded as an almost everywhere function, exactly as the $L^{p}$ classes of Complex Lp classes and Euclidean test-function conventions are.
One writes $∥ S f ∥_{p}$ for the $L^{p}$ norm of the class of $S f$ when $S f \in L^{p}$ ; the norm is that of $L^{p} (R^{n}; C)$ , with $S f$ interpreted as the complex-valued function $x \mapsto S f (x)$ (it is real-valued and nonnegative). The notation $S_{F} f$ for a finite set $F \subset {0, 1, 2, \dots}$ means $(\sum_{j \in F} ∣ Δ_{j} f ∣^{2})^{1 / 2}$ , so that $S_{N} f = S_{{0, \dots, N - 1}} f$ .

The operator-theoretic counterpart of $S$ for functions on the frequency side is not asserted here: the definition names only the pointwise square function of the convolution representatives, and the strict-range equivalence with $∥ f ∥_{p}$ is a theorem proved later on this page.

LemmaStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Rademacher randomisation turns dyadic square functions into random signed multipliers

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). For every $0 < p < \infty$ there are constants $0 < c_{p} \leq C_{p} < \infty$ depending only on $p$ with the following property. For every finite set $F \subset {0, 1, 2, \dots}$ , every $f \in S (R^{n})$ and every $x \in R^{n}$ , $c_{p} (\sum_{j \in F} ∣ Δ_{j} f (x) ∣^{2})^{1 / 2} \leq (\int_{0}^{1} ∣ \sum_{j \in F} ε_{j} (t) Δ_{j} f (x) ∣^{p} d t)^{1 / p} \leq C_{p} (\sum_{j \in F} ∣ Δ_{j} f (x) ∣^{2})^{1 / 2} .$ Moreover, for every fixed $t \in [0, 1)$ the finite sum $\sum_{j \in F} ε_{j} (t) Δ_{j} f$ equals the Fourier multiplier $T_{m_{t, F}} f$ with symbol $m_{t, F} : = \sum_{j \in F} ε_{j} (t) φ_{j} \in C_{c}^{\infty} (R^{n})$ ; integrating the pointwise inequality in $x$ and using Tonelli gives, for every $f \in S$ , $c_{p}^{p} ∥ S_{F} f ∥_{p}^{p} \leq \int_{0}^{1} ∥ T_{m_{t, F}} f ∥_{p}^{p} d t \leq C_{p}^{p} ∥ S_{F} f ∥_{p}^{p} .$ The same statements hold with $Δ_{j}$ replaced by the companion operators ${\tilde{Δ}}_{j}$ .

Facts & Assumptions

Given: $0 < p < \infty$ , a finite set $F \subset {0, 1, 2, \dots}$ , a Schwartz function $f \in S (R^{n})$ and a point $x \in R^{n}$ ; the fixed partition $(φ_{j})$ , operators $Δ_{j}$ and kernels $K_{j}$ of The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators; the square function $S_{F} f = (\sum_{j \in F} ∣ Δ_{j} f ∣^{2})^{1 / 2}$ of The Littlewood-Paley square function.

[F1]

Khintchine's inequality: for every finite sequence $(a_{j})_{j \in J}$ of complex numbers and every $0 < p < \infty$ there are $0 < c_{p} \leq C_{p} < \infty$ , depending only on $p$ , with $c_{p} (\sum_{j} ∣ a_{j} ∣^{2})^{1 / 2} \leq (\int_{0}^{1} ∣ \sum_{j} ε_{j} (t) a_{j} ∣^{p} d t)^{1 / p} \leq C_{p} (\sum_{j} ∣ a_{j} ∣^{2})^{1 / 2}$ , and the constants are independent of $J$ (Khintchine's inequality for finite Rademacher sums, Rademacher functions on the unit interval).

[F2]

For $f \in S$ the operators act as $Δ_{j} f = T_{φ_{j}} f = F^{- 1} (φ_{j} \hat{f})$ , and $T$ is linear in the symbol: for smooth compactly supported symbols $m, n$ one has $T_{m} + T_{n} = T_{m + n}$ on $S$ . The finite sum $m_{t, F} : = \sum_{j \in F} ε_{j} (t) φ_{j}$ lies in $C_{c}^{\infty} (R^{n})$ ; if $F \neq \emptyset$ , writing $j_{F} : = \max F$ , its support is contained in ${∣ ξ ∣ \leq 2^{j_{F} + 2}}$ , while for $F = \emptyset$ it is the zero symbol with empty support (The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators). For $f \in S$ , $T_{m_{t, F}} f$ is a Schwartz function and the multiplier action is convolution by the corresponding kernel.

[F3]

$(t, x) \mapsto \sum_{j \in F} ε_{j} (t) Δ_{j} f (x)$ is measurable on $[0, 1) \times R^{n}$ : it is a finite sum of products of Borel functions of $t$ with continuous functions of $x$ (Arithmetic and lattice operations preserve measurability whenever they are defined); hence $∣ \cdot ∣^{p}$ of it is nonnegative and product-measurable, and Tonelli's theorem applies, in particular $\int_{0}^{1} \int_{R^{n}} ∣ \cdot ∣^{p} d x d t = \int_{R^{n}} \int_{0}^{1} ∣ \cdot ∣^{p} d t d x$ (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product).

Proof

technique · direct

1.1F1algebra

Pointwise Khintchine. At the fixed point $x$ apply [F1] to the finite coefficient vector $a_{j} : = Δ_{j} f (x)$ , $j \in F$ (legitimate because the coefficients are complex numbers); this gives exactly the displayed pointwise two-sided inequality, with constants depending only on $p$ and not on $F$ , $f$ or $x$ .

1.2F2algebra

The random sum is a multiplier. Fix $t \in [0, 1)$ . By [F2] each $Δ_{j} f = T_{φ_{j}} f$ is in $S$ and the multiplier map is linear in its symbol, so $\sum_{j \in F} ε_{j} (t) Δ_{j} f = T_{m_{t, F}} f$ with $m_{t, F} = \sum_{j \in F} ε_{j} (t) φ_{j} \in C_{c}^{\infty} (R^{n})$ ; in particular $T_{m_{t, F}} f \in S$ and $\int_{R^{n}} ∣ T_{m_{t, F}} f ∣^{p} d x < \infty$ .

2.1F3step 1.1step 1.2algebra

The integrated inequality. By [F3] the function $∣ \sum_{j \in F} ε_{j} (t) Δ_{j} f (x) ∣^{p}$ is nonnegative and product-measurable, and $S_{F} f$ is measurable by The Littlewood-Paley square function; the pointwise inequality of step 1.1 passes to the $x$ -integral, so $c_{p}^{p} ∥ S_{F} f ∥_{p}^{p} \leq \int_{R^{n}} \int_{0}^{1} ∣ \sum_{j} ε_{j} (t) Δ_{j} f (x) ∣^{p} d t d x \leq C_{p}^{p} ∥ S_{F} f ∥_{p}^{p}$ . Since the inner expression equals $∣ T_{m_{t, F}} f (x) ∣^{p}$ by step 1.2, Tonelli [F3] rewrites the middle term as $\int_{0}^{1} ∥ T_{m_{t, F}} f ∥_{p}^{p} d t$ ; the quantities are finite because a finite sign vector takes only finitely many values, each corresponding random sum is Schwartz, and a finite sum of Schwartz moduli lies in $L^{p}$ for every $p > 0$ by choosing decay exponent $M$ with $M p > n$ .

3.1F1F2step 1.1step 2.1algebra∎

Companions. Replacing $K_{j}$ by ${\tilde{K}}_{j}$ , $φ_{j}$ by ${\tilde{φ}}_{j}$ and $Δ_{j}$ by ${\tilde{Δ}}_{j}$ throughout, steps 1.1 to 2.1 apply verbatim with $S_{F}$ replaced by ${\tilde{S}}_{F} f = (\sum_{j \in F} ∣ {\tilde{Δ}}_{j} f ∣^{2})^{1 / 2}$ , because the companion symbols are again compactly supported smooth functions and the Khintchine inequality is insensitive to the coefficients.

LemmaStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Random signed dyadic sums have uniform Mihlin and Lp multiplier bounds

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). For every sequence $(c_{j})_{j \geq 0}$ of complex numbers with $∣ c_{j} ∣ \leq 1$ the symbol $m : = \sum_{j \geq 0} c_{j} φ_{j}$ of The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators is a Mihlin symbol with constants $C_{α} \leq A_{α} (n, ψ)$ for $∣ α ∣ \leq ⌊ n / 2 ⌋ + 1$ , where $A_{α}$ does not depend on the coefficients and the series is locally finite. Consequently, for every $1 < p < \infty$ and every $f \in L^{p} (R^{n}; C)$ , $∥ T_{m} f ∥_{p} \leq C_{n, ψ} \max (p, (p - 1)^{- 1}) ∥ f ∥_{p}$ with $C_{n, ψ}$ independent of the coefficients. In particular the bound is uniform over all sign sequences $c_{j} = \pm 1$ and over all finite truncations, that is over $\sum_{j < N} ε_{j} φ_{j}$ for every $N$ and every choice of signs.

Facts & Assumptions

Given: the fixed partition $(φ_{j})$ of The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators; a sequence $(c_{j})$ with $∣ c_{j} ∣ \leq 1$ ; $q : = ⌊ n / 2 ⌋ + 1$ ; a real $1 < p < \infty$ .

[F1]

Each $φ_{j} \in C_{c}^{\infty} (R^{n})$ satisfies $0 \leq φ_{j} \leq 1$ , $\sum_{j} φ_{j} = 1$ with a locally finite sum, $supp φ_{j} \subset {2^{j - 1} \leq ∣ ξ ∣ \leq 2^{j + 1}}$ for $j \geq 1$ and $supp φ_{0} \subset {∣ ξ ∣ \leq 2}$ ; for every multi-index $α$ there is $C_{α} < \infty$ with $∣ \partial^{α} φ_{j} (ξ) ∣ \leq C_{α} 2^{- j ∣ α ∣}$ for $j \geq 1$ and all $ξ$ , with $∣ \partial^{α} φ_{0} (ξ) ∣ \leq C_{α}$ (Existence of a smooth inhomogeneous dyadic frequency partition, The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators).

[F2]

Mihlin's theorem: if $m$ is a Mihlin symbol with constants $C_{α}$ and $A : = \max_{∣ α ∣ \leq q} C_{α}$ , then $m$ is an $L^{p}$ Fourier multiplier and $∥ m ∥_{M_{p}} \leq C_{n} \max (p, (p - 1)^{- 1}) (A + ∥ m ∥_{\infty})$ (Mihlin smoothness convention above half the dimension, The Mihlin–Hörmander Fourier multiplier theorem, Lp Fourier multiplier and its norm, Translation-invariant Fourier multiplier on the Schwartz core).

Proof

technique · direct

1.1F1algebra

The symbol is well defined and bounded. At each fixed $ξ$ at most three of the numbers $φ_{j} (ξ)$ are nonzero by [F1], so the series $m (ξ) = \sum_{j} c_{j} φ_{j} (ξ)$ is actually a finite sum at every point and converges locally uniformly to a smooth function; and $∣ m (ξ) ∣ \leq \sum_{j} φ_{j} (ξ) = 1$ because $∣ c_{j} ∣ \leq 1$ and $φ_{j} \geq 0$ .

2.1F1step 1.1algebra

Uniform Mihlin constants. For $α = 0$ step 1.1 gives $∣ m ∣ \leq 1$ , which is the required degree-zero bound with constant $1$ . Fix $α$ with $1 \leq ∣ α ∣ \leq q$ and $ξ \neq 0$ , and put $k : = ∣ α ∣$ . By [F1], $∣ \partial^{α} m (ξ) ∣ \leq \sum_{j \geq 0} ∣ \partial^{α} φ_{j} (ξ) ∣$ . If the $j = 0$ derivative is nonzero, then $∣ ξ ∣ \leq 2$ , so $∣ ξ ∣^{k} ∣ \partial^{α} φ_{0} (ξ) ∣ \leq 2^{k} C_{α}$ . For $j \geq 1$ , a nonzero derivative requires $2^{j - 1} \leq ∣ ξ ∣ \leq 2^{j + 1}$ ; at most three integers $j$ satisfy both inequalities. On each such annulus, [F1] gives $∣ ξ ∣^{k} ∣ \partial^{α} φ_{j} (ξ) ∣ \leq ∣ ξ ∣^{k} C_{α} 2^{- j k} \leq 2^{k} C_{α}$ . Therefore $∣ ξ ∣^{k} ∣ \partial^{α} m (ξ) ∣ \leq 4 \cdot 2^{k} C_{α}$ , which is the required homogeneous Mihlin estimate. Thus $m$ is a Mihlin symbol with constants $A_{0} : = 1$ and $A_{α} : = 4 \cdot 2^{∣ α ∣} C_{α}$ for $1 \leq ∣ α ∣ \leq q$ , depending only on $n, ψ, α$ , uniformly in the coefficients.

3.1F2step 1.1step 2.1algebra

Uniform $L^{p}$ bounds. By step 2.1 the symbol $m$ is Mihlin with $A : = \max_{∣ α ∣ \leq q} A_{α}$ and, by step 1.1, $∥ m ∥_{\infty} \leq 1$ ; [F2] therefore gives the $L^{p}$ multiplier bound $∥ T_{m} f ∥_{p} \leq C_{n} \max (p, (p - 1)^{- 1}) (A + 1) ∥ f ∥_{p}$ for every $f \in L^{p}$ , with constants depending only on $n, ψ, p$ .

4.1step 3.1algebra∎

Truncations and signs. A finite truncation $\sum_{j < N} ε_{j} φ_{j}$ is the symbol $m$ for the sequence $c_{j} = ε_{j} 1_{j < N}$ , which again satisfies $∣ c_{j} ∣ \leq 1$ ; the bound of step 3.1 therefore applies to all such truncations and to all sign sequences $c_{j} = \pm 1$ , with the same constant $C_{n, ψ}$ .

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The Littlewood-Paley reproducing formula in tempered distributions

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). With the fixed partition and companion operators of The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators:

for every $f \in S^{'} (R^{n})$ the partial sums $\sum_{j < N} {\tilde{Δ}}_{j} Δ_{j} f$ converge to $f$ in $S^{'} (R^{n})$ as $N \to \infty$ , that is $f = \sum_{j \geq 0} {\tilde{Δ}}_{j} Δ_{j} f$ in $S^{'}$ ;
for every $f, g \in S (R^{n})$ and every $N \geq 0$ , with the Hermitian pairing $⟨ u, v ⟩ = \int_{R^{n}} u \overset{ˉ}{v}$ one has $⟨ \sum_{j < N} {\tilde{Δ}}_{j} Δ_{j} f, g ⟩ = \sum_{j < N} ⟨ Δ_{j} f, {\tilde{Δ}}_{j} g ⟩$ . If in addition $∥ S f ∥_{p} < \infty$ and $∥ \tilde{S} g ∥_{p^{'}} < \infty$ for some $1 < p < \infty$ , with $p^{'}$ conjugate to $p$ , then the series $\sum_{j \geq 0} ⟨ Δ_{j} f, {\tilde{Δ}}_{j} g ⟩$ converges absolutely to $⟨ f, g ⟩$ and satisfies $\sum_{j \geq 0} ∣ ⟨ Δ_{j} f, {\tilde{Δ}}_{j} g ⟩ ∣ \leq \int_{R^{n}} S f \tilde{S} g$ .

Facts & Assumptions

Given: the fixed partition $(φ_{j})$ , companions $({\tilde{φ}}_{j})$ and operators of The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators; the partial symbols $σ_{N} : = \sum_{j < N} {\tilde{φ}}_{j} φ_{j}$ for $N \geq 0$ .

[F1]

Each $φ_{j}, {\tilde{φ}}_{j}$ lies in $C_{c}^{\infty} (R^{n})$ , $0 \leq φ_{j} \leq 1$ , $0 \leq {\tilde{φ}}_{j} \leq 1$ and $∣ \partial^{α} φ_{j} (ξ) ∣ \leq C_{α} 2^{- j ∣ α ∣}$ , $∣ \partial^{α} {\tilde{φ}}_{j} (ξ) ∣ \leq C_{α}^{'} 2^{- j ∣ α ∣}$ for constants depending only on $n, ψ, α$ ; the supports satisfy $supp φ_{j} \subset {2^{j - 1} \leq ∣ ξ ∣ \leq 2^{j + 1}}$ for $j \geq 1$ and $supp {\tilde{φ}}_{j} \subset {∣ ξ ∣ \leq 2^{j + 2}}$ (Existence of a smooth inhomogeneous dyadic frequency partition, The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators).

[F2]

$\sum_{j \geq 0} {\tilde{φ}}_{j} φ_{j} = 1$ pointwise with a locally finite sum, and the operators satisfy $T_{m} T_{n} = T_{m n}$ on $S$ for smooth polynomially bounded symbols; for $f \in S^{'}$ the products $φ_{j} \cdot F f$ are the transposed multiplications by smooth polynomially bounded symbols, and $Δ_{j} f = F^{- 1} (φ_{j} F f)$ (The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators, Smooth polynomially bounded multipliers on schwartz space).

[F3]

The distributional pairing is bilinear, with $⟨ F f, g ⟩ = ⟨ f, F g ⟩$ and $⟨ F^{- 1} f, g ⟩ = ⟨ f, F^{- 1} g ⟩$ . $F$ is an automorphism of $S^{'} (R^{n})$ and $F^{- 1} F = F F^{- 1} = i d$ ; for $f \in S^{'}$ the map $g \mapsto ⟨ f, g ⟩$ is a continuous linear functional on $S$ (Fourier transform of a tempered distribution, Fourier transform is a topological automorphism of tempered distributions, Fourier transform is a topological automorphism of Schwartz space, Schwartz topology and convergence, Weak and strong topologies on tempered distributions).

[F4]

Parseval's pairing: for $u, v \in S (R^{n})$ , $\int \hat{u} \overline{\hat{v}} = \int u \overset{ˉ}{v}$ ; consequently, for a real symbol $m \in C_{c}^{\infty}$ and $u, v \in S$ , writing $\hat{T_{m} u} = m \hat{u}$ , $⟨ T_{m} u, v ⟩ = \int \hat{T_{m} u} \overset{ˉ}{\hat{v}} = \int m \hat{u} \overset{ˉ}{\hat{v}} = \int \hat{u} \overline{m \hat{v}} = ⟨ u, T_{m} v ⟩$ (Parseval pairing on Schwartz space, The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators).

[F5]

For $h \in S$ and $R > 0$ , $\sup_{∣ ξ ∣ \geq R} ∣ ξ^{α} \partial^{β} h (ξ) ∣ \leq R^{- 1} \sum_{l = 1}^{n} p_{α + e_{l}, β} (h)$ : multiply by $∣ ξ ∣ \leq \sum_{l} ∣ ξ_{l} ∣$ and use the seminorm bounds. Thus the tail tends to $0$ , uniformly when $h$ ranges over a bounded subset of $S$ ; a sequence converges in $S$ exactly when all seminorms tend to $0$ (Schwartz topology and convergence).

[F6]

Holder's inequality and the monotone convergence theorem for nonnegative measurable functions; finite sums act termwise on integrals (Complex Holder, Minkowski, and the quotient norm, Monotone convergence for the integral).

Proof

technique · direct

1.1F1F2algebra

The partial symbols. For every $N \geq 0$ the function $σ_{N} = \sum_{j < N} {\tilde{φ}}_{j} φ_{j}$ lies in $C_{c}^{\infty} (R^{n})$ by [F1] and satisfies $0 \leq σ_{N} \leq 1$ with $1 - σ_{N} = \sum_{j \geq N} {\tilde{φ}}_{j} φ_{j} \geq 0$ by [F2]. For $N \geq 1$ it equals $1$ on ${∣ ξ ∣ < 2^{N - 2}}$ : a term ${\tilde{φ}}_{j} φ_{j}$ with $j \geq N$ vanishes wherever $φ_{j}$ does, and $φ_{j} = 0$ on ${∣ ξ ∣ \leq 2^{j - 1}}$ , so every tail term vanishes on this ball. (For $N = 0$ , $σ_{0} = 0$ and no plateau assertion is made.) Moreover, for every multi-index $α$ there is $C_{α} < \infty$ , depending only on $n, ψ, α$ , with $∣ \partial^{α} (1 - σ_{N}) (ξ) ∣ \leq C_{α} 2^{- N ∣ α ∣}$ for all $ξ$ and $N$ : by the Leibniz rule and [F1] each summand satisfies $∣ \partial^{α} ({\tilde{φ}}_{j} φ_{j}) (ξ) ∣ \leq C_{α} 2^{- j ∣ α ∣}$ , and $\sum_{j \geq N} 2^{- j ∣ α ∣} \leq 2 \cdot 2^{- N ∣ α ∣}$ for $∣ α ∣ \geq 1$ , while for $α = 0$ the bound is just $0 \leq 1 - σ_{N} \leq 1$ .

1.2F4F6algebra

The finite duality identity and absolute convergence. For $f, g \in S$ and $N$ , [F4] applied to the real symbol ${\tilde{φ}}_{j}$ gives $⟨ {\tilde{Δ}}_{j} Δ_{j} f, g ⟩ = ⟨ Δ_{j} f, {\tilde{Δ}}_{j} g ⟩$ for each $j < N$ , and summing yields the first identity of part 2. For the series bound, the pointwise Cauchy-Schwarz inequality in $ℓ^{2}$ gives $\sum_{j < N} ∣ Δ_{j} f (x) ∣ ∣ {\tilde{Δ}}_{j} g (x) ∣ \leq S f (x) \tilde{S} g (x)$ for every $x$ , and integrating the finite sum, which is legitimate termwise by [F6], gives $\sum_{j < N} \int ∣ Δ_{j} f ∣ ∣ {\tilde{Δ}}_{j} g ∣ \leq \int S f \tilde{S} g \leq ∥ S f ∥_{p} ∥ \tilde{S} g ∥_{p^{'}} < \infty$ by Holder; hence the nonnegative series $\sum_{j} \int ∣ Δ_{j} f ∣ ∣ {\tilde{Δ}}_{j} g ∣$ converges (its partial sums are increasing and bounded) and dominates $\sum_{j} ∣ ⟨ Δ_{j} f, {\tilde{Δ}}_{j} g ⟩ ∣$ , so that series converges absolutely with the stated bound.

2.1F1F5step 1.1algebra

The tail vanishes in $S$ . For every $h \in S$ the products $(1 - σ_{N}) h$ tend to $0$ in the Schwartz topology: for multi-indices $α, β$ , the Leibniz rule writes $\partial^{β} ((1 - σ_{N}) h) = \sum_{γ \leq β} (\binom{β}{γ}) \partial^{β - γ} (1 - σ_{N}) \partial^{γ} h$ , and every term with $γ \neq β$ is bounded by $C_{β - γ} 2^{- N ∣ β - γ ∣} ∣ ξ^{α} \partial^{γ} h (ξ) ∣ \leq C 2^{- N} p_{α γ} (h) \to 0$ uniformly in $ξ$ , while, for $N \geq 1$ , the term with $γ = β$ satisfies $∣ ξ^{α} (1 - σ_{N}) \partial^{β} h (ξ) ∣ \leq \sup_{∣ ξ ∣ \geq 2^{N - 2}} ∣ ξ^{α} \partial^{β} h (ξ) ∣ \to 0$ by [F5], since $1 - σ_{N}$ vanishes for $∣ ξ ∣ < 2^{N - 2}$ . Hence $p_{α β} ((1 - σ_{N}) h) \to 0$ for every pair $α, β$ , which is convergence to $0$ in $S$ by [F5], uniformly on bounded subsets because the finitely many seminorms in these estimates are uniformly bounded.

3.1F2F3step 2.1algebra

Convergence in $S^{'}$ (part 1). The multiplier composition in [F2] gives $\sum_{j < N} {\tilde{Δ}}_{j} Δ_{j} f = F^{- 1} (σ_{N} F f)$ for every $f \in S^{'}$ . For a Schwartz test $g$ , the bilinear transposition convention [F3] gives $⟨ F^{- 1} (σ_{N} F f) - f, g ⟩ = ⟨ F f, (σ_{N} - 1) F^{- 1} g ⟩$ . By step 2.1 the test on the right tends to zero in $S$ , and continuity of $F f$ makes the pairing tend to zero. The same estimates are uniform for $g$ in any bounded subset: $F^{- 1}$ maps bounded sets to bounded sets, step 2.1 is uniform there, and a continuous functional is bounded by finitely many Schwartz seminorms. Hence the partial sums converge to $f$ in both the weak and strong dual topologies.

4.1step 1.2step 3.1∎

Identification of the sum (part 2). Apply step 3.1 to the Schwartz test $\overset{ˉ}{g}$ ; the resulting distributional pairing is the Hermitian integral pairing of part 2. Thus $⟨ \sum_{j < N} {\tilde{Δ}}_{j} Δ_{j} f, g ⟩ \to ⟨ f, g ⟩$ in that convention. Step 1.2 identifies each partial sum with $\sum_{j < N} ⟨ Δ_{j} f, {\tilde{Δ}}_{j} g ⟩$ and proves absolute convergence with the stated bound, so the sum is $⟨ f, g ⟩$ .

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Littlewood-Paley square-function equivalence on Lp for 1<p<infinity

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $n \geq 1$ , fix the partition and operators of The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators, and let $S$ be the square function of The Littlewood-Paley square function.

For every $1 < p < \infty$ there are constants $0 < c_{p} \leq C_{p} < \infty$ , depending only on $n$ , $p$ and the fixed cutoff, such that every $f \in S (R^{n})$ satisfies $c_{p} ∥ f ∥_{L^{p}} \leq ∥ S f ∥_{L^{p}} \leq C_{p} ∥ f ∥_{L^{p}};$ in particular $S f \in L^{p}$ for Schwartz $f$ .
(Extension to $L^{p}$ .) For every $f \in L^{p} (R^{n}; C)$ and every sequence $f_{k} \in S (R^{n})$ with $f_{k} \to f$ in $L^{p}$ , the sequence $S f_{k}$ is Cauchy in $L^{p}$ ; its limit, written $S f$ , is independent of the approximating sequence, satisfies the same two-sided estimate with the constants of part 1, and is the unique continuous extension of the Schwartz assignment. Moreover the increasing sequence $S_{N} f = (\sum_{j < N} ∣ Δ_{j} f ∣^{2})^{1 / 2}$ of convolution representatives converges to $S f$ in $L^{p}$ and almost everywhere, so $S f = (\sum_{j \geq 0} ∣ Δ_{j} f ∣^{2})^{1 / 2}$ almost everywhere.

Only $1 < p < \infty$ is claimed.

Facts & Assumptions

Given: the fixed partition, operators and kernels of The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators; the square function $S$ and its truncations $S_{N}$ , $S_{F}$ of The Littlewood-Paley square function; a real $1 < p < \infty$ with conjugate $p^{'}$ ; the finite partial symbols $m_{t, N} = \sum_{j < N} ε_{j} (t) φ_{j}$ for $t \in [0, 1)$ .

[F1]

Rademacher randomisation: for every $0 < q < \infty$ there are $0 < c_{q} \leq C_{q} < \infty$ , depending only on $q$ , such that for every finite $F$ , every $f \in S$ and every $x$ , $c_{q} S_{F} f (x) \leq (\int_{0}^{1} ∣ \sum_{j \in F} ε_{j} (t) Δ_{j} f (x) ∣^{q} d t)^{1 / q} \leq C_{q} S_{F} f (x)$ , and for each $t$ the random sum equals $T_{m_{t, F}} f$ ; integrating in $x$ gives $c_{q}^{q} ∥ S_{F} f ∥_{q}^{q} \leq \int_{0}^{1} ∥ T_{m_{t, F}} f ∥_{q}^{q} d t \leq C_{q}^{q} ∥ S_{F} f ∥_{q}^{q}$ (Rademacher randomisation turns dyadic square functions into random signed multipliers, Khintchine's inequality for finite Rademacher sums).

[F2]

Uniform signed-sum bounds: for every sequence $∣ c_{j} ∣ \leq 1$ the symbol $\sum_{j} c_{j} φ_{j}$ is a Mihlin symbol with constants independent of the coefficients, and for $1 < q < \infty$ one has $∥ T_{\sum_{j} c_{j} φ_{j}} f ∥_{q} \leq C_{n, ψ} \max (q, (q - 1)^{- 1}) ∥ f ∥_{q}$ ; in particular the bound applies to every truncation symbol $m_{t, N}$ and is uniform in $t, N$ (Random signed dyadic sums have uniform Mihlin and Lp multiplier bounds, Dyadic pieces are uniformly Mihlin multipliers and uniformly Lp-bounded).

[F3]

Reproducing formula: for $f, g \in S$ with $∥ S f ∥_{p} < \infty$ and $∥ \tilde{S} g ∥_{p^{'}} < \infty$ the series $\sum_{j} ⟨ Δ_{j} f, {\tilde{Δ}}_{j} g ⟩$ converges absolutely to $⟨ f, g ⟩$ and $\sum_{j} ∣ ⟨ Δ_{j} f, {\tilde{Δ}}_{j} g ⟩ ∣ \leq \int S f \tilde{S} g$ (The Littlewood-Paley reproducing formula in tempered distributions).

[F4]

Holder's inequality and the finite $ℓ^{2}$ reverse-triangle inequality $∣ S_{N} u - S_{N} v ∣ \leq S_{N} (u - v)$ ; its limit gives $∣ S u - S v ∣ \leq S (u - v)$ wherever $S u$ and $S v$ are finite (Complex Holder, Minkowski, and the quotient norm, The Littlewood-Paley square function).

[F5]

Integration and limit tools: Tonelli for nonnegative product-measurable integrands, monotone convergence for increasing sequences, dominated convergence in $L^{p}$ , completeness of $L^{p}$ together with almost everywhere convergence of a subsequence, density of smooth compactly supported functions in $L^{p}$ for finite $p$ , and the duality formula $∥ h ∥_{p} = \sup {∣ \int h g ∣ : ∥ g ∥_{p^{'}} \leq 1}$ (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, Monotone convergence for the integral, Dominated convergence, Complex Lp completeness and almost-everywhere subsequences, Complex finite-simple and smooth compact-support density for finite p, The $L^{p}$ norm is the supremum of pairings against unit $L^{q}$ functions).

Proof

technique · direct

1.1F1F2F5algebra

Upper bound for Schwartz functions. Let $f \in S$ and $N \geq 0$ . The lower pointwise Khintchine bound of [F1] with $q = p$ gives $S_{N} f (x) \leq c_{p}^{- 1} (\int_{0}^{1} ∣ \sum_{j < N} ε_{j} (t) Δ_{j} f (x) ∣^{p} d t)^{1 / p}$ ; raising to the $p$ -th power, integrating in $x$ , and applying Tonelli (the integrand is nonnegative and product-measurable by [F1]) gives $∥ S_{N} f ∥_{p}^{p} \leq c_{p}^{- p} \int_{0}^{1} ∥ T_{m_{t, N}} f ∥_{p}^{p} d t \leq c_{p}^{- p} (C_{n, ψ} \max (p, (p - 1)^{- 1}))^{p} ∥ f ∥_{p}^{p}$ , where the last inequality is the uniform signed-sum bound [F2] applied at each $t$ to the truncation symbols. Since $S_{N} f ↑ S f$ pointwise, monotone convergence [F5] gives $∥ S f ∥_{p} \leq C_{p} ∥ f ∥_{p}$ with $C_{p} : = c_{p}^{- 1} C_{n, ψ} \max (p, (p - 1)^{- 1})$ ; in particular $S f \in L^{p}$ . The same computation for any finite set $F$ in place of ${0, \dots, N - 1}$ gives $∥ S_{F} f ∥_{p} \leq C_{p} ∥ f ∥_{p}$ .

2.1F5step 1.1algebra

Companion upper bound. Since ${\tilde{φ}}_{j} = φ_{j - 1} + φ_{j} + φ_{j + 1}$ and the multiplier map is linear in the symbol, ${\tilde{Δ}}_{j} f = Δ_{j - 1} f + Δ_{j} f + Δ_{j + 1} f$ (with $Δ_{- 1} : = 0$ ); hence $∣ {\tilde{Δ}}_{j} f ∣^{2} \leq 3 (∣ Δ_{j - 1} f ∣^{2} + ∣ Δ_{j} f ∣^{2} + ∣ Δ_{j + 1} f ∣^{2})$ pointwise, and summing over $j \in F$ gives ${\tilde{S}}_{F} f \leq 3 S_{F^{'}} f$ with the finite set $F^{'} = {j - 1, j, j + 1 : j \in F} \cap {0, 1, 2, \dots}$ . By step 1.1 applied to $F^{'}$ , $∥ {\tilde{S}}_{F} f ∥_{p} \leq 3 C_{p} ∥ f ∥_{p}$ ; letting $F$ increase to all indices and using monotone convergence for ${\tilde{S}}_{F} f ↑ \tilde{S} f$ gives $∥ \tilde{S} f ∥_{p} \leq 3 C_{p} ∥ f ∥_{p}$ .

2.2F4F5step 1.1algebra

Extension to $L^{p}$ . For $u, v \in S$ the pointwise inequality $∣ S u - S v ∣ \leq S (u - v)$ of [F4] and step 1.1 give $∥ S u - S v ∥_{p} \leq C_{p} ∥ u - v ∥_{p}$ ; hence $u \mapsto S u$ is Lipschitz on the dense subspace $S$ of $L^{p}$ , and for any $f \in L^{p}$ and any $f_{k} \in S$ with $f_{k} \to f$ the sequence $S f_{k}$ is Cauchy in $L^{p}$ . By completeness of $L^{p}$ [F5] it has a limit $F$ , which is independent of the approximating sequence: if ${\tilde{f}}_{k}$ is another such sequence, the interleaved sequence $f_{1}, {\tilde{f}}_{1}, f_{2}, {\tilde{f}}_{2}, \dots$ also converges to $f$ in $L^{p}$ and its image is Cauchy, so the two limits agree. This assignment is the unique continuous extension of the Schwartz square function, by the same Lipschitz bound.

3.1F3F4F5step 1.1step 2.1algebra

Lower bound for Schwartz functions. Let $f \in S$ . By steps 1.1 and 2.1, $∥ S f ∥_{p} < \infty$ and, for every $g \in S$ , $∥ \tilde{S} g ∥_{p^{'}} < \infty$ ; the reproducing formula [F3] therefore gives $⟨ f, g ⟩ = \sum_{j} ⟨ Δ_{j} f, {\tilde{Δ}}_{j} g ⟩$ with $\sum_{j} ∣ ⟨ Δ_{j} f, {\tilde{Δ}}_{j} g ⟩ ∣ \leq \int S f \tilde{S} g$ . By the pointwise Cauchy-Schwarz inequality and Holder [F4], $\int S f \tilde{S} g \leq ∥ S f ∥_{p} ∥ \tilde{S} g ∥_{p^{'}} \leq 3 C_{p^{'}} ∥ S f ∥_{p} ∥ g ∥_{p^{'}}$ . Hence $∣ ⟨ f, g ⟩ ∣ \leq 3 C_{p^{'}} ∥ S f ∥_{p} ∥ g ∥_{p^{'}}$ first for all $g \in S$ and then, by density of $S$ in $L^{p^{'}}$ and continuity of the pairing, for all $g \in L^{p^{'}}$ ; taking the supremum over $∥ g ∥_{p^{'}} \leq 1$ and using the duality formula [F5] gives $∥ f ∥_{p} \leq 3 C_{p^{'}} ∥ S f ∥_{p}$ . This is the lower bound of part 1 with $c_{p} : = (3 C_{p^{'}})^{- 1}$ .

4.1F2F4F5step 1.1step 2.2step 3.1algebra

Identification with the pointwise square function. For fixed $N$ , [F2] and the finite reverse-triangle inequality give $∥ S_{N} u - S_{N} v ∥_{p} \leq \sum_{j < N} ∥ Δ_{j} (u - v) ∥_{p} \leq N B_{p} ∥ u - v ∥_{p}$ for all $u, v \in L^{p}$ , where $B_{p}$ is a uniform bound for the pieces. Therefore $S_{N}$ is continuous on $L^{p}$ , and approximation by Schwartz functions passes the bound of step 1.1 to $∥ S_{N} f ∥_{p} \leq C_{p} ∥ f ∥_{p}$ for every $f \in L^{p}$ , uniformly in $N$ . Monotone convergence [F5] gives $∥ S f ∥_{p} \leq C_{p} ∥ f ∥_{p}$ for the increasing pointwise limit, so $S f$ is finite almost everywhere. Since $∣ S f - S_{N} f ∣^{p} \leq (S f)^{p}$ , dominated convergence [F5] gives $S_{N} f \to S f$ in $L^{p}$ , as well as pointwise. Passing the finite reverse-triangle inequality to the limit gives $∣ S u - S v ∣ \leq S (u - v)$ almost everywhere and hence $∥ S u - S v ∥_{p} \leq C_{p} ∥ u - v ∥_{p}$ on all of $L^{p}$ . In particular, for $f_{k} \in S$ with $f_{k} \to f$ , $S f_{k} \to S f$ in $L^{p}$ , identifying this function with the extension of step 2.2. Taking limits in steps 1.1 and 3.1 gives the two-sided estimate for $f$ .

5.1step 1.1step 2.1step 2.2step 3.1step 4.1∎

Conclusion. Steps 1.1 and 3.1 prove part 1, and steps 2.2 and 4.1 prove part 2: the Cauchy property and independence of the approximating sequence, the identification of the extension with the increasing pointwise square function both in $L^{p}$ and almost everywhere, and the inherited two-sided estimate.

CorollaryStatement: Literature-sourcedProof: AI-adaptedOpen item page →

The choice of admissible dyadic partition does not change the Lp square-function space

Statement

Assume Countable Choice. Call an inhomogeneous dyadic frequency partition admissible when it is obtained as in Existence of a smooth inhomogeneous dyadic frequency partition from some radial cutoff satisfying the standing hypotheses. Let $(φ_{j})$ and $(ψ_{j})$ be two admissible partitions with associated square functions $S^{(φ)}$ and $S^{(ψ)}$ . Then for every $1 < p < \infty$ there are constants $0 < c \leq C < \infty$ , depending only on $n, p$ and the two cutoffs, such that for every $f \in L^{p} (R^{n}; C)$ $c ∥ S^{(ψ)} f ∥_{p} \leq ∥ S^{(φ)} f ∥_{p} \leq C ∥ S^{(ψ)} f ∥_{p} .$ Moreover the mixed pieces are almost orthogonal: $Δ_{j}^{(φ)} Δ_{k}^{(ψ)} = 0$ whenever $∣ j - k ∣ \geq 3$ , and each mixed operator $Δ_{j}^{(φ)} Δ_{k}^{(ψ)}$ is the Fourier multiplier with symbol $φ_{j} ψ_{k}$ supported in $supp φ_{j} \cap supp ψ_{k}$ . For $j, k \geq 1$ these are intersections of the corresponding annular supports; if either index is zero, use the actual low-frequency ball support of that block.

Facts & Assumptions

Given: Countable Choice, two admissible partitions $(φ_{j})$ , $(ψ_{j})$ with cutoffs $ψ_{φ}, ψ_{ψ}$ and operators $Δ_{j}^{(φ)}, Δ_{k}^{(ψ)}$ ; a real $1 < p < \infty$ ; a function $f \in L^{p} (R^{n}; C)$ .

[F1]

Littlewood-Paley square-function equivalence on Lp for 1<p<infinity applies to each admissible partition separately: for the partition $(φ_{j})$ there are constants $0 < c_{1} \leq C_{1} < \infty$ depending only on $n, p$ and the cutoff $ψ_{φ}$ with $c_{1} ∥ g ∥_{p} \leq ∥ S^{(φ)} g ∥_{p} \leq C_{1} ∥ g ∥_{p}$ for all $g \in L^{p}$ , and for the partition $(ψ_{j})$ there are constants $0 < c_{2} \leq C_{2} < \infty$ depending only on $n, p$ and $ψ_{ψ}$ with $c_{2} ∥ g ∥_{p} \leq ∥ S^{(ψ)} g ∥_{p} \leq C_{2} ∥ g ∥_{p}$ .

[F2]

For each of the two admissible partitions, every fixed dyadic piece is a bounded operator on $L^{p}$ and agrees with its convolution representative; the composition on $S$ has symbol $φ_{j} ψ_{k}$ (Dyadic pieces are uniformly Mihlin multipliers and uniformly Lp-bounded, The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators). Also $C_{c}^{\infty}$ is dense in $L^{p}$ for $1 < p < \infty$ ( $C_{c}^{\infty} (R^{n})$ is dense in $L^{p} (R^{n})$ for $1 \leq p < \infty$ ). For $j, k \geq 1$ the supports satisfy $supp φ_{j} \subset {2^{j - 1} \leq ∣ ξ ∣ \leq 2^{j + 1}}$ and $supp ψ_{k} \subset {2^{k - 1} \leq ∣ ξ ∣ \leq 2^{k + 1}}$ ; the low blocks satisfy $supp φ_{0} \subset {∣ ξ ∣ \leq 2}$ and $supp ψ_{0} \subset {∣ ξ ∣ \leq 2}$ (The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators, Existence of a smooth inhomogeneous dyadic frequency partition).

Proof

technique · direct

1.1F1algebra

Comparability of the two square functions. By [F1] applied to $f$ , $c_{1} ∥ f ∥_{p} \leq ∥ S^{(φ)} f ∥_{p} \leq C_{1} ∥ f ∥_{p}$ and $c_{2} ∥ f ∥_{p} \leq ∥ S^{(ψ)} f ∥_{p} \leq C_{2} ∥ f ∥_{p}$ ; eliminating $∥ f ∥_{p}$ gives $\frac{c_{1}}{C_{2}} ∥ S^{(ψ)} f ∥_{p} \leq ∥ S^{(φ)} f ∥_{p} \leq \frac{C_{1}}{c_{2}} ∥ S^{(ψ)} f ∥_{p}$ , with constants depending only on $n, p$ and the two cutoffs.

1.2F2algebra

The mixed pieces and their supports. For $f \in S$ the composition rule [F2] gives $Δ_{j}^{(φ)} Δ_{k}^{(ψ)} f = T_{φ_{j} ψ_{k}} f$ , whose symbol is supported in $supp φ_{j} \cap supp ψ_{k}$ . If $∣ j - k ∣ \geq 3$ , say $k \geq j + 3$ , then $supp φ_{j} \subset {∣ ξ ∣ \leq 2^{j + 1}}$ (including $j = 0$ , since its support lies in the radius-two ball) and $supp ψ_{k} \subset {∣ ξ ∣ \geq 2^{k - 1}}$ , with $2^{j + 1} < 2^{k - 1}$ , so the supports are disjoint and $φ_{j} ψ_{k} \equiv 0$ . Thus the mixed operator vanishes on $S$ ; by [F2] each fixed dyadic piece is bounded on $L^{p}$ , so its composition is bounded, and $C_{c}^{\infty} \subset S$ is dense in $L^{p}$ . Hence the identity $Δ_{j}^{(φ)} Δ_{k}^{(ψ)} = 0$ extends to all of $L^{p}$ . The same argument applies when $j \geq k + 3$ after interchanging the partitions. When $j, k \geq 1$ the support intersection is the intersection of their annuli; if a low block occurs it is the intersection with that block's actual ball support from [F2].

2.1step 1.1step 1.2∎

Conclusion. Step 1.1 is the stated two-sided comparability with constants depending only on $n, p$ and the two cutoffs, and step 1.2 is the almost-orthogonality and support statement for the mixed pieces.

TheoremStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Littlewood-Paley characterisation of the Hilbert-Sobolev spaces

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $n \geq 1$ and $s \in R$ , and let $H^{s} (R^{n})$ be the real-order Bessel-potential space with the exact norm $∥ U ∥_{H^{s}} = ∥ ⟨ ξ ⟩^{s} F (E_{s} U) ∥_{2}$ of Real-order H^s as weighted Fourier distributions, where $⟨ ξ ⟩ = (1 + ∣ ξ ∣^{2})^{1 / 2}$ . Fix the partition of The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators and let $Δ_{j} U$ be the tempered distributions obtained by the multipliers $φ_{j}$ . Then for every $U \in S^{'} (R^{n})$ , $U$ lies in the image of the canonical embedding $E_{s}$ (and is identified with its preimage in $H^{s}$ ) if and only if $\sum_{j \geq 0} 2^{2 j s} ∥ Δ_{j} U ∥_{L^{2}}^{2} < \infty,$ where $∥ Δ_{j} U ∥_{L^{2}}$ denotes the $L^{2}$ norm of the unique $L^{2}$ function representing the tempered distribution $Δ_{j} U$ when such a function exists and is set equal to $+ \infty$ otherwise (the convention is needed only for the converse direction; for $U$ in the image of $E_{s}$ every $Δ_{j} U$ is a regular $L^{2}$ distribution, as the proof records), and in that case $c_{n, s, ψ} ∥ U ∥_{H^{s}}^{2} \leq \sum_{j \geq 0} 2^{2 j s} ∥ Δ_{j} U ∥_{L^{2}}^{2} \leq C_{n, s, ψ} ∥ U ∥_{H^{s}}^{2}$ with constants depending only on $n, s$ and the partition. The low-frequency block carries the weight $2^{0} = 1$ , and the series converges absolutely.

Facts & Assumptions

Given: $n \geq 1$ , $s \in R$ , the completion $H^{s} (R^{n})$ of $S$ with norm $q_{s}$ , its canonical embedding $E_{s} : H^{s} \to S^{'}$ and the isometry $J_{s}$ of The Bessel completion embeds canonically in tempered distributions; the fixed partition $(φ_{j})$ and operators $Δ_{j}$ of The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators; a tempered distribution $U \in S^{'} (R^{n})$ .

[F1]

$H^{s}$ is the normed completion of $S$ under $q_{s} (u) = ∥ ⟨ ξ ⟩^{s} \hat{u} ∥_{2}$ , $E_{s}$ is the canonical embedding and $J_{s} ([u_{j}]) = \lim_{j} ⟨ ξ ⟩^{s} F u_{j}$ is a surjective linear isometry $H^{s} \to L^{2}$ (Real-order Bessel-potential completion H^s, The Bessel completion embeds canonically in tempered distributions); the multiplier $⟨ D ⟩^{t}$ and the bracket powers are those of Japanese-bracket and Laplacian Bessel-potential operators and Real powers of the Japanese bracket act on Schwartz space.

[F2]

Characterisation: $E_{s}$ is a bijection from $H^{s}$ onto the set of $U \in S^{'}$ for which $⟨ ξ ⟩^{s} F U = u_{g}$ for some $g \in L^{2}$ , the class $g$ is unique, and then $∥ U ∥_{H^{s}} = ∥ g ∥_{2}$ (Real-order H^s as weighted Fourier distributions).

[F3]

For $U \in S^{'}$ , $Δ_{j} U = F^{- 1} (φ_{j} F U)$ ; each $φ_{j} \in C_{c}^{\infty}$ , $0 \leq φ_{j} \leq 1$ , $supp φ_{j} \subset {2^{j - 1} \leq ∣ ξ ∣ \leq 2^{j + 1}}$ for $j \geq 1$ and $supp φ_{0} \subset {∣ ξ ∣ \leq 2}$ , the sum $\sum_{j} φ_{j}^{2}$ lies in $[1 / 3, 1]$ pointwise with at most three nonzero terms, and $\sum_{j} φ_{j} = 1$ (The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators, Existence of a smooth inhomogeneous dyadic frequency partition).

[F4]

Plancherel: $F_{2}$ and $F_{2}^{- 1}$ are isometries of $L^{2}$ , so $∥ g ∥_{2} = ∥ F_{2} g ∥_{2}$ ; if a tempered distribution is the regular distribution $u_{H}$ of an $L^{2}$ function $H$ , its representing $L^{2}$ class is unique and $∥ H ∥_{2}$ is the corresponding norm (Plancherel theorem, Exact L2 Fourier multiplier norm).

[F5]

The support bounds [F3] give $2^{j - 1} \leq ⟨ ξ ⟩ \leq \sqrt{5} 2^{j}$ for $j \geq 1$ and $1 \leq ⟨ ξ ⟩ \leq \sqrt{5}$ for $j = 0$ . Thus the bracket and the dyadic scale are comparable: $⟨ ξ ⟩ ≍_{n} 2^{j}$ for $j \geq 1$ , and $⟨ ξ ⟩ ≍_{n} 1$ together with $2^{0} = 1$ for $j = 0$ ; hence $c_{s} \leq 2^{2 j s} ⟨ ξ ⟩^{- 2 s} \leq C_{s}$ on each $supp φ_{j}$ with constants depending only on $n$ and $s$ (this also covers negative $s$ , since the comparison is two-sided) (Japanese-bracket and Laplacian Bessel-potential operators).

[F6]

Holder's inequality and Cauchy-Schwarz for integrals and finite sums (Complex Holder, Minkowski, and the quotient norm).

[F7]

Tonelli's theorem permits interchanging the nonnegative sums and integrals used below (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product).

[F8]

Fourier transformation of a regular $L^{2}$ distribution agrees with Plancherel (Fourier transform agrees with l one and plancherel transforms); locally integrable densities are determined almost everywhere by their distribution pairings (Locally integrable functions embed in distributions). $C_{c}^{\infty}$ is dense in $S$ (Smooth compact supports are dense in Schwartz space), and $L^{2}$ is complete under Countable Choice (Complex Lp completeness and almost-everywhere subsequences).

Proof

technique · direct

1.1F1F2F3F4F8algebra

Forward direction: identification of the pieces. Let $U = E_{s} (V)$ with $V \in H^{s}$ and let $g = J_{s} (V) \in L^{2}$ , so that $∥ U ∥_{H^{s}} = ∥ g ∥_{2}$ and $⟨ ξ ⟩^{s} F U = u_{g}$ by [F2]. Put $h : = ⟨ ξ ⟩^{- s} g$ ; since $⟨ ξ ⟩^{- s}$ is smooth and locally bounded, $h \in L_{l o c}^{2} (R^{n})$ , and $u_{h} = ⟨ ξ ⟩^{- s} u_{g} = F U$ by the invertibility of the bracket multiplier [F1]. Then $Δ_{j} U = F^{- 1} (φ_{j} F U) = F^{- 1} (u_{φ_{j} h}) = u_{(φ_{j} h)^{\lor}}$ by [F3], and $φ_{j} h \in L^{2}$ because $h$ is locally square integrable and $φ_{j}$ is compactly supported and bounded; Plancherel [F4] gives $∥ Δ_{j} U ∥_{2}^{2} = ∥ φ_{j} h ∥_{2}^{2} = \int_{R^{n}} φ_{j} (ξ)^{2} ∣ h (ξ) ∣^{2} d ξ$ .

1.2F3F4F5F6F8algebra

Converse direction: reconstruction. Assume $\sum_{j} 2^{2 j s} ∥ Δ_{j} U ∥_{2}^{2} < \infty$ ; then for every $j$ the distribution $Δ_{j} U$ is the regular distribution $u_{H_{j}}$ of an $L^{2}$ function $H_{j}$ (the case $H_{j} = 0$ included), and we may take $H_{j} = Δ_{j} U$ as an $L^{2}$ class. Put $h_{j} : = F_{2} H_{j}$ and $g_{j} : = ⟨ ξ ⟩^{s} h_{j}$ . Since $F (Δ_{j} U) = φ_{j} F U$ and $F u_{H_{j}} = u_{F_{2} H_{j}}$ , the distribution $u_{h_{j}} = φ_{j} F U$ is supported in $supp φ_{j}$ , so $h_{j} = 0$ almost everywhere off that support; by [F5] this gives $∥ g_{j} ∥_{2}^{2} = \int ⟨ ξ ⟩^{2 s} ∣ h_{j} ∣^{2} ≍_{n, s} 2^{2 j s} ∥ H_{j} ∥_{2}^{2} = 2^{2 j s} ∥ Δ_{j} U ∥_{2}^{2}$ , so $\sum_{j} ∥ g_{j} ∥_{2}^{2} < \infty$ . The supports of the $g_{j}$ lie in the supports of the $φ_{j}$ , at most three of which meet at any point by [F3]; hence $∣ \sum_{j \in F} g_{j} ∣^{2} \leq 3 \sum_{j \in F} ∣ g_{j} ∣^{2}$ pointwise for every finite $F$ . Therefore the partial sums are Cauchy in $L^{2}$ (their tail squared norms are at most three times the tails of $\sum_{j} ∥ g_{j} ∥_{2}^{2}$ ) and converge by [F8] to some $g \in L^{2}$ , with $∥ g ∥_{2}^{2} \leq 3 \sum_{j} ∥ g_{j} ∥_{2}^{2} ≍_{n, s} \sum_{j} 2^{2 j s} ∥ Δ_{j} U ∥_{2}^{2}$ . No lower norm estimate is used until the reconstruction identifies $g_{j} = φ_{j} g$ .

2.1F3F5F7step 1.1algebra

Forward direction: the weighted sum. Multiplying the identity of step 1.1 by $2^{2 j s}$ and summing, the pointwise comparison $2^{2 j s} ⟨ ξ ⟩^{- 2 s} \in [c_{s}, C_{s}]$ on $supp φ_{j}$ from [F5] gives $\sum_{j \geq 0} 2^{2 j s} ∥ Δ_{j} U ∥_{2}^{2} = \sum_{j} \int 2^{2 j s} φ_{j}^{2} ∣ h ∣^{2} ≍_{n, s} \int (\sum_{j} φ_{j}^{2}) ∣ g ∣^{2}$ , where [F7] interchanges the nonnegative sum and integral and the last comparison uses $∣ h ∣^{2} = ⟨ ξ ⟩^{- 2 s} ∣ g ∣^{2}$ . Since $\sum_{j} φ_{j}^{2} \in [1 / 3, 1]$ by [F3], this is comparable to $\int ∣ g ∣^{2} = ∥ U ∥_{H^{s}}^{2}$ ; in particular the series is finite and the right-hand inequality with $C_{n, s, ψ}$ holds, while the left-hand inequality follows from the same comparison with $c_{s}$ .

2.2F2F3F6F7F8step 1.2algebra

Converse direction: $U$ lies in the range of $E_{s}$ . With $g$ as in step 1.2, test against any $χ \in C_{c}^{\infty}$ . Only finitely many $φ_{j}$ meet its compact support, and $\sum_{j} φ_{j} = 1$ there by [F3], so $⟨ ⟨ ξ ⟩^{s} F U, χ ⟩ = \sum_{j} \int g_{j} χ = \int g χ$ , the last equality following from the $L^{2}$ convergence in step 1.2 and Cauchy-Schwarz [F6]. Both sides are tempered distributions, and density of $C_{c}^{\infty}$ in $S$ [F8] extends this identity to every Schwartz test, giving $⟨ ξ ⟩^{s} F U = u_{g}$ . Multiplication by $φ_{j}$ then shows $g_{j} = φ_{j} g$ as $L^{2}$ functions. Tonelli [F7] and [F3] give $\frac{1}{3} ∥ g ∥_{2}^{2} \leq \sum_{j} ∥ g_{j} ∥_{2}^{2} = \int (\sum_{j} φ_{j} (ξ)^{2}) ∣ g (ξ) ∣^{2} d ξ \leq ∥ g ∥_{2}^{2} .$ Thus $U$ lies in the range of $E_{s}$ by [F2], and this frame comparison, the piecewise comparison in step 1.2, and the exact norm identity $∥ U ∥_{H^{s}} = ∥ g ∥_{2}$ give $c_{n, s, ψ} \sum_{j} 2^{2 j s} ∥ Δ_{j} U ∥_{2}^{2} \leq ∥ U ∥_{H^{s}}^{2} \leq C_{n, s, ψ} \sum_{j} 2^{2 j s} ∥ Δ_{j} U ∥_{2}^{2} .$

3.1step 2.1step 2.2∎

Conclusion. Steps 2.1 and 2.2 are the two directions of the asserted equivalence and the two-sided norm comparison (the constants of step 2.1 for the forward direction and those of step 2.2 for the converse are both of the form $c_{n, s, ψ}, C_{n, s, ψ}$ ); the weight $2^{0} = 1$ on the low-frequency block is part of the definition of the series, and the finiteness of the series in the forward direction is contained in step 2.1.

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

The Lusin area function for a fixed admissible kernel and aperture

Definition

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Fix an aperture $a > 0$ and an admissible kernel: a real-valued radial Schwartz function $ψ$ (Schwartz space and its seminorms) with $\int_{R^{n}} ψ = 0$ and $ψ$ not identically zero; the support convention of The support of a function on $R^{n}$ and its compactly supported Riemann integral applies to the compactly supported functions used below.

For $t > 0$ write $ψ_{t} (y) : = t^{- n} ψ (y / t)$ , and for $x \in R^{n}$ let $Γ_{a} (x) : = {(y, t) \in R^{n} \times (0, \infty) : ∣ y - x ∣ < a t}$ be the cone of aperture $a$ over $x$ . For $f \in L^{p} (R^{n}; C)$ with $1 \leq p \leq \infty$ , or for $f \in S (R^{n})$ (which lies in every $L^{p}$ ), fix a representative of $f$ and define $A_{a, ψ} f (x) : = (\int_{0}^{\infty} \int_{∣ y - x ∣ < a t} ∣ (f * ψ_{t}) (y) ∣^{2} d y \frac{d t}{t^{n + 1}})^{1 / 2} \in [0, \infty],$ where $f * ψ_{t}$ is the convolution of Convolution of two functions on $R^{n}$ . The following well-definedness facts are part of the definition and are used with the cited suppliers.

Let $q$ be conjugate to $p$ , with $q = \infty$ when $p = 1$ and $q = 1$ when $p = \infty$ . For every $y \in R^{n}$ and $t > 0$ , Holder's inequality (Complex Holder, Minkowski, and the quotient norm) makes $(f * ψ_{t}) (y) : = \int_{R^{n}} f (u) t^{- n} ψ ((y - u) / t) d u$ absolutely convergent and independent of the representative of $f$ . Young's inequality (Young's convolution inequality under Countable Choice) gives $∥ f * ψ_{t} ∥_{p} \leq ∥ ψ ∥_{1} ∥ f ∥_{p}$ . Moreover $(y, t) \mapsto t^{- n} ψ ((y - \cdot) / t)$ is continuous into $L^{q} (R^{n})$ for $t > 0$ : near any fixed $(y, t)$ these Schwartz kernels depend pointwise continuously on the parameters and have a common integrable Schwartz majorant for finite $q$ , so dominated convergence applies (Dominated convergence); for $q = \infty$ , uniform continuity on bounded sets and a uniform Schwartz tail give convergence in the supremum norm. Holder's inequality therefore shows that $(y, t) \mapsto (f * ψ_{t}) (y)$ is jointly continuous, hence Borel measurable.
For each $x$ the set $Γ_{a} (x)$ is open in $R^{n} \times (0, \infty)$ and the integrand is nonnegative, so the iterated integral over $Γ_{a} (x)$ is well defined in $[0, \infty]$ by Tonelli (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product) without any integrability hypothesis; the comparison with the integral over the measurable set $Γ_{a} (x)$ is the convention of Integral over a measurable subset.
$A_{a, ψ} f$ is Borel measurable as a function of $x$ , in fact it is lower semicontinuous. Write $H (y, t) : = ∣ (f * ψ_{t}) (y) ∣^{2}$ ; this is continuous by item 1. If $x_{m} \to x$ , then $1_{{∣ y - x ∣ < a t}} \leq {lim inf}_{m} 1_{{∣ y - x_{m} ∣ < a t}}$ for every $(y, t)$ , because the cone inequality is strict. Fatou's lemma (Fatou's lemma) applied to the nonnegative integrands with measure $d y d t / t^{n + 1}$ gives $A_{a, ψ} f (x)^{2} \leq {lim inf}_{m} A_{a, ψ} f (x_{m})^{2}$ ; hence the extended-valued function $A_{a, ψ} f$ is lower semicontinuous and therefore Borel.
If two representatives of $f$ agree almost everywhere, their integrands in the integral formula of item 1 agree almost everywhere in $u$ ; the Lebesgue integral respects almost-everywhere equality (Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree), so the convolutions, and hence the area functions, agree at every $(y, t)$ and $x$ . Thus the functional is defined on almost-everywhere classes, with values allowed to equal $+ \infty$ .

The cancellation $\int ψ = 0$ and the aperture $a$ are part of the data. Distinct pairs need not give distinct functionals: replacing $ψ$ by $- ψ$ leaves $A_{a, ψ}$ unchanged, since the squared modulus of every convolution is unchanged. No equivalence between this functional and a square-function scale is asserted here.

RemarkRemark: Literature-sourcedProof: Not applicableOpen item page →

Littlewood-Paley endpoint scope and the H1-BMO dual pair

Statement

The two-sided square-function equivalence of Littlewood-Paley square-function equivalence on Lp for 1<p<infinity is stated for $1 < p < \infty$ . Its contract supplies no estimate at $p = 1$ or $p = \infty$ , so substituting either endpoint into it or its consumers is not justified by that theorem.

The locally defined real Hardy space $H^{1} (R^{n})$ is The real Hardy space $H^{p}$ defined by a radial maximal function. Under the Axiom of Choice, its continuous dual is isomorphic, with equivalent norms, to BMO modulo constants (BMO seminorm and the quotient by constants, Real H1-BMO duality), with the fixed Hardy kernel and auxiliary order required by that duality theorem. Thus the library supplies the pair $(H^{1}, B M O / C)$ as a proved duality of these defined spaces. That duality supplies no square-function endpoint estimate on its own.

Choice. The dual identification inherits the full Axiom of Choice from Real H1-BMO duality, declared through The Axiom of Choice. The strict-range statement is used only within its own hypotheses. The conclusions here use the defined spaces and local duality; no homogeneous or inhomogeneous endpoint square-function characterization is asserted.

5 · Examples, counterexamples and false statements

None yet.

Littlewood Paley Theory and Square Functions

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

Rademacher functions on the unit interval

Definition

Finite Rademacher blocks are equidistributed

Statement

Facts & Assumptions

Proof

Khintchine's inequality for finite Rademacher sums

Statement

Facts & Assumptions

Proof

Existence of a smooth inhomogeneous dyadic frequency partition

Statement

Facts & Assumptions

Proof

The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators

Definition

Dyadic pieces have annular Fourier support and uniformly bounded rescaled kernels

Statement

Facts & Assumptions

Proof

Dyadic pieces are uniformly Mihlin multipliers and uniformly Lp-bounded

Statement

Facts & Assumptions

Proof

L2 almost orthogonality of the dyadic pieces

Statement

Facts & Assumptions

Proof

The Littlewood-Paley square function

Definition

Rademacher randomisation turns dyadic square functions into random signed multipliers

Statement

Facts & Assumptions

Proof

Random signed dyadic sums have uniform Mihlin and Lp multiplier bounds

Statement

Facts & Assumptions

Proof

The Littlewood-Paley reproducing formula in tempered distributions

Statement

Facts & Assumptions

Proof

Littlewood-Paley square-function equivalence on Lp for 1<p<infinity

Statement

Facts & Assumptions

Proof

The choice of admissible dyadic partition does not change the Lp square-function space

Statement

Facts & Assumptions

Proof

Littlewood-Paley characterisation of the Hilbert-Sobolev spaces

Statement

Facts & Assumptions

Proof

The Lusin area function for a fixed admissible kernel and aperture

Definition

Littlewood-Paley endpoint scope and the H1-BMO dual pair

Statement

5 · Examples, counterexamples and false statements

Sources