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The Riesz transforms are bounded on Lp

Statement

Assume Countable Choice. Let $n \geq 1$ and $1 \leq j \leq n$ . The $j$ -th Riesz transform $R_{j}$ of Riesz transforms on Euclidean space extends uniquely to a bounded operator on $L^{p} (R^{n}; C)$ for every $1 < p < \infty$ , with norm at most $C_{n, p}$ , a constant depending only on $n$ and $p$ : the Riesz kernel $K_{j} (x) = c_{n} x_{j} / ∣ x ∣^{n + 1}$ is a standard $1$ -Hölder Calderón–Zygmund kernel with constant $C_{n} = c_{n} 2^{n + 1} (3 n + 4)$ , so that $∥ R_{j} f ∥_{p} \leq C_{n, p} (1 + ∣ S^{n - 1} ∣ 2^{- 1} C_{n}) \max (p, (p - 1)^{- 1}) ∥ f ∥_{p} (f \in L^{p} (R^{n}; C)) .$

Facts & Assumptions

Given: Countable Choice; the dimension $n \geq 1$ and index $1 \leq j \leq n$ ; the Riesz kernel $K_{j} (x) = c_{n} x_{j} / ∣ x ∣^{n + 1}$ and operator $R_{j}$ of Riesz transforms on Euclidean space; a compactly supported $f \in L^{2} (R^{n}; C)$ ; a test function $φ \in C_{c}^{\infty} (R^{n})$ supported off $supp f$ .

[F1]

$∣ K_{j} (x) ∣ \leq c_{n} ∣ x ∣^{- n}$ for $x \neq 0$ ; $∣ K_{j} (x - h) - K_{j} (x) ∣ \leq C_{n} ∣ h ∣ ∣ x ∣^{- (n + 1)}$ with $C_{n} = c_{n} 2^{n + 1} (3 n + 4)$ whenever $∣ h ∣ \leq ∣ x ∣ / 2$ ; and $\int_{S^{n - 1}} K_{j} (r ω) d σ (ω) = 0$ for every $r > 0$ (Riesz kernel size, difference and spherical-cancellation bounds).

[F2]

For every Schwartz function $g$ the truncated integrals $\int_{∣ y ∣ > ε} K_{j} (y) g (x - y) d y$ converge as $ε ↓ 0$ for every $x$ , and the limit is a continuous representative of the $L^{2}$ class $R_{j} g$ (The Riesz transform is the principal value of its kernel, with the matching constant).

[F3]

$R_{j}$ is the $L^{2}$ Fourier multiplier with symbol $m_{j} (ξ) = - i ξ_{j} / ∣ ξ ∣$ for $ξ \neq 0$ and $m_{j} (0) = 0$ , is bounded with $∥ R_{j} g ∥_{2} \leq ∥ g ∥_{2}$ for all $g \in L^{2}$ , and satisfies $⟨ R_{j} f, g ⟩ = \int m_{j} \hat{f} \overline{\hat{g}}$ for the first-variable-linear pairing $⟨ u, v ⟩ = \int u \overline{v}$ (Riesz transforms on Euclidean space, Riesz transforms are L2 contractions and square to minus the identity in sum, Plancherel theorem, The Hilbert-space adjoint of a bounded operator).

[F4]

A pointwise bound $∣ k ∣ \leq c ∣ \cdot ∣^{- n}$ implies the annular condition with $A_{1} = c ∣ S^{n - 1} ∣ \log 2$ ; a standard $δ$ -Hölder kernel with constant $A_{2}^{'}$ is a Calderón–Zygmund kernel with Hörmander constant $A_{2} = ∣ S^{n - 1} ∣ 2^{- δ} δ^{- 1} A_{2}^{'}$ ; and a Calderón–Zygmund operator with constants $A_{1}, A_{2}$ and $L^{2}$ norm $B$ extends uniquely to a bounded operator on $L^{p}$ for $1 < p < \infty$ with $∥ T g ∥_{p} \leq C_{n, p} (A_{2} + B) \max (p, (p - 1)^{- 1}) ∥ g ∥_{p}$ (Calderón–Zygmund kernels and their associated operators, Standard (Hölder) Calderón–Zygmund kernels, Standard Hölder kernels satisfy the Hörmander condition, Calderón–Zygmund operators are bounded on Lp).

[F5]

Fubini interchanges absolutely integrable complex double integrals, and locally integrable functions with equal distribution pairings agree almost everywhere. (Fubini's theorem for L^1 functions on a sigma-finite product, Locally integrable functions embed in distributions)

[F6]

Under Countable Choice, for every nonnegative Borel function $g$ , $\int_{R^{n}} g (x) d x = \int_{0}^{\infty} \int_{S^{n - 1}} g (r ω) r^{n - 1} d σ (ω) d r$ , with $σ$ a finite Borel measure. (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma)

Proof

technique · direct

1.1F1given

The published estimates give the size bound $∣ K_{j} (x) ∣ \leq c_{n} ∣ x ∣^{- n}$ , the first-difference bound $∣ K_{j} (x - h) - K_{j} (x) ∣ \leq C_{n} ∣ h ∣ ∣ x ∣^{- (n + 1)}$ for $∣ h ∣ \leq ∣ x ∣ / 2$ , and the vanishing of every spherical mean. The explicit kernel is smooth on the punctured space, hence Borel measurable and locally integrable there.

1.2F3givenalgebra

$R_{j}$ is $L^{2}$ -bounded with $∥ R_{j} g ∥_{2} \leq ∥ g ∥_{2}$ , since $∣ m_{j} ∣ \leq 1$ and the $L^{2}$ multiplier bound gives $∥ R_{j} ∥ \leq 1$ ; hence $B = 1$ is an admissible $L^{2}$ norm bound. Moreover $R_{j}$ is skew-adjoint: for $f, g \in L^{2}$ , $⟨ R_{j} f, g ⟩ = \int m_{j} \hat{f} \overline{\hat{g}} = \int \hat{f} \overline{(- m_{j}) \hat{g}} = ⟨ f, - R_{j} g ⟩$ , because $\overline{m_{j}} = - m_{j}$ for the purely imaginary symbol; that is, $R_{j}^{*} = - R_{j}$ .

2.1F1F4F6step 1.1algebra

By [F6], the size bound gives $\int_{R \leq ∣ x ∣ \leq 2 R} ∣ K_{j} (x) ∣ d x \leq c_{n} ∣ S^{n - 1} ∣ \int_{R}^{2 R} d r / r = c_{n} ∣ S^{n - 1} ∣ \log 2$ . For every $h \neq 0$ , directly integrating the difference bound gives $\int_{∣ x ∣ \geq 2 ∣ h ∣} ∣ K_{j} (x - h) - K_{j} (x) ∣ d x \leq C_{n} ∣ h ∣ ∣ S^{n - 1} ∣ \int_{2 ∣ h ∣}^{\infty} r^{- 2} d r = ∣ S^{n - 1} ∣ C_{n} / 2$ . Thus $K_{j}$ is a base Calderón–Zygmund kernel with $A_{1} = c_{n} ∣ S^{n - 1} ∣ \log 2$ and $A_{2} = ∣ S^{n - 1} ∣ C_{n} / 2$ ; its first-difference estimate now establishes standard $1$ -Hölder status with $A_{2}^{'} = C_{n}$ .

2.2F2F3step 1.2algebraF5

Off-support representation: let $f \in L^{2}$ be compactly supported, let $φ \in C_{c}^{\infty}$ be supported off $supp f$ , and put $d : = dist (supp f, supp φ) > 0$ . Using the adjoint identity of [F3], skew-adjointness from step 1.2 and the Schwartz principal-value formula [F2], and writing $K_{j}$ for the real-valued kernel, $⟨ R_{j} f, φ ⟩ = ⟨ f, R_{j}^{*} φ ⟩ = - \int f (y) \overline{R_{j} φ (y)} d y = - \iint K_{j} (y - x) f (y) \overline{φ (x)} d x d y = \iint K_{j} (x - y) f (y) \overline{φ (x)} d x d y,$ where $\overline{R_{j} φ (y)} = \lim_{ε ↓ 0} \int_{∣ y - x ∣ > ε} K_{j} (y - x) \overline{φ (x)} d x$ is an absolutely convergent integral on the two supports (there $∣ x - y ∣ \geq d > 0$ , so the limit may be taken inside the $y$ -integration), Fubini applies over the bounded supports, and the last step uses the oddness $K_{j} (- z) = - K_{j} (z)$ . Since this holds for every test function supported off $supp f$ , the $L^{2}$ class $R_{j} f$ agrees almost everywhere off $supp f$ with the locally integrable function $x \mapsto \int K_{j} (x - y) f (y) d y$ ; this is the off-support representation (3) required of a Calderón–Zygmund operator.

3.1F4step 2.1step 1.2step 2.2∎

By steps 2.1, 1.2 and 2.2 the operator $R_{j}$ is a standard-kernel Calderón–Zygmund operator with annular constant $c_{n} ∣ S^{n - 1} ∣ \log 2$ , Hörmander constant $∣ S^{n - 1} ∣ 2^{- 1} C_{n}$ and $L^{2}$ norm $B = 1$ ; the strict-range theorem [F4] therefore gives its unique extension to a bounded operator on $L^{p} (R^{n}; C)$ , $1 < p < \infty$ , with $∥ R_{j} f ∥_{p} \leq C_{n, p} (1 + ∣ S^{n - 1} ∣ 2^{- 1} C_{n}) \max (p, (p - 1)^{- 1}) ∥ f ∥_{p}$ . This is the assertion.

Depends on

Used by

Almost-everywhere convergence of principal-value truncations Corollary

Dependency tree · two levels

81 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

Loukas Grafakos, Classical Fourier Analysis, third edition (standard reference, not scraped)
Richard S. Laugesen, Harmonic Analysis Lecture Notes (standard reference, not scraped)