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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.

Depends on

Used by

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

Dependency tree · two levels

59 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Terence Tao, Math 247A Lecture Notes 4 (UCLA, Fall 2006) (standard reference, not scraped)
Mark Williams, Notes on Harmonic Analysis (January 11, 2022) (standard reference, not scraped)
Loukas Grafakos, Classical Fourier Analysis, 3rd ed. (Springer GTM 249) (standard reference, not scraped)