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The Riesz kernel is a standard Calderón–Zygmund kernel

Example

Assume Countable Choice. For $n \geq 1$ and $1 \leq j \leq n$ the Riesz kernel $K_{j} (x) = c_{n} x_{j} / ∣ x ∣^{n + 1}$ satisfies the pointwise size bound $∣ K_{j} (x) ∣ \leq c_{n} ∣ x ∣^{- n}$ , the $δ = 1$ first-difference bound on $∣ x ∣ \geq 2 ∣ y ∣ > 0$ , and the zero spherical-mean estimate; the Riesz transform $R_{j}$ is $L^{2}$ -bounded and the published principal-value formula identifies its truncations on Schwartz functions. Hence $R_{j}$ is a standard-kernel Calderón–Zygmund operator in the sense of this page.

Facts & Assumptions

Given: Countable Choice; the integer $n \geq 1$ and index $1 \leq j \leq n$ ; a compactly supported $f \in L^{2}$ and a test function $φ \in C_{c}^{\infty}$ supported off $supp f$ .

[F1]

$K_{j} (x) = c_{n} x_{j} / ∣ x ∣^{n + 1}$ satisfies $∣ K_{j} (x) ∣ \leq c_{n} ∣ x ∣^{- n}$ ; $∣ K_{j} (x - h) - K_{j} (x) ∣ \leq C_{n} ∣ h ∣ ∣ x ∣^{- (n + 1)}$ whenever $x \neq 0$ and $∣ h ∣ \leq ∣ x ∣ / 2$ ; 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 $ε ↓ 0$ to a continuous representative of the $L^{2}$ class $R_{j} g$ , for every $x$ (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 $ξ \neq 0$ , $∥ R_{j} f ∥_{2} \leq ∥ f ∥_{2}$ , and Plancherel's theorem identifies the pairing $⟨ R_{j} f, g ⟩$ with $\int m_{j} \hat{f} \overline{\hat{g}}$ for the first-variable-linear pairing (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 Calderón–Zygmund kernel is a measurable $k$ on $R^{n} ∖ {0}$ , locally integrable there, with finite annular constant and finite Hörmander constant; a pointwise bound $∣ k ∣ \leq c ∣ \cdot ∣^{- n}$ implies the annular condition with $A_{1} = c ∣ S^{n - 1} ∣ \log 2$ ; a Calderón–Zygmund operator is an $L^{2}$ -bounded linear operator satisfying the off-support kernel representation for compactly supported $L^{2}$ inputs (Calderón–Zygmund kernels and their associated operators, Standard (Hölder) Calderón–Zygmund kernels, Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma). Polar coordinates give $\int_{∣ x ∣ \geq a} ∣ x ∣^{- n - 1} d x = ∣ S^{n - 1} ∣ / a$ for $a > 0$ .

[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)

Verification

technique · direct

1.1F1F3given

The published estimates give exactly the size bound $∣ K_{j} (x) ∣ \leq c_{n} ∣ x ∣^{- n}$ , the $δ = 1$ first-difference bound in the regime $∣ x ∣ \geq 2 ∣ h ∣ > 0$ with constant $A_{2}^{'} = C_{n}$ , and the vanishing of all spherical means of $K_{j}$ . The formula for $K_{j}$ makes it smooth, hence measurable and locally integrable, off the origin.

1.2F3given

$R_{j}$ is $L^{2}$ -bounded with norm at most one, since its symbol $m_{j}$ is measurable with $∣ m_{j} ∣ \leq 1$ on ${ξ \neq 0}$ and the Plancherel multiplier bound gives $∥ R_{j} ∥ \leq ∥ m_{j} ∥_{\infty} \leq 1$ ; thus $B = 1$ is an admissible $L^{2}$ norm bound.

1.3F3givenalgebra

$R_{j}$ 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; hence $R_{j}^{*} = - R_{j}$ .

2.1F1F4step 1.1algebra

The pointwise size bound gives the annular condition with $A_{1} = c_{n} ∣ S^{n - 1} ∣ \log 2$ . For every $h \neq 0$ , integrating the raw difference bound in [F1] and using polar coordinates yields $\int_{∣ x ∣ \geq 2 ∣ h ∣} ∣ K_{j} (x - h) - K_{j} (x) ∣ d x \leq C_{n} ∣ h ∣ \int_{∣ x ∣ \geq 2 ∣ h ∣} ∣ x ∣^{- n - 1} d x = C_{n} ∣ h ∣ \frac{∣ S^{n - 1} ∣}{2 ∣ h ∣} = \frac{1}{2} C_{n} ∣ S^{n - 1} ∣ .$ Thus Hörmander's condition holds with $A_{2} = ∣ S^{n - 1} ∣ 2^{- 1} C_{n}$ . Together with step 1.1 this proves that $K_{j}$ is a Calderón–Zygmund kernel, and its raw first-difference bound now makes it standard $1$ -Hölder with constant $C_{n}$ .

2.2F1F2F3step 1.3algebraF5

Let $f \in L^{2}$ have compact support and let $φ \in C_{c}^{\infty}$ be supported off it. If either is zero the formula is immediate; otherwise the supports have distance $d > 0$ . By skew-adjointness and [F2], $\int (R_{j} f) φ = ⟨ R_{j} f, \overline{φ} ⟩ = - \int f (y) \overline{R_{j} \overline{φ} (y)} d y = - \iint K_{j} (y - x) f (y) φ (x) d x d y = \iint K_{j} (x - y) f (y) φ (x) d x d y$ . The kernel is real and odd, and the positive separation and bounded supports make the double integral absolutely convergent. Fubini and injectivity of locally integrable distribution pairings identify $R_{j} f$ almost everywhere off its support with $\int K_{j} (x - y) f (y) d y$ .

3.1F4step 1.1step 2.1step 1.2step 2.2∎

Steps 2.1, 1.2 and 2.2 verify the annular and Hörmander bounds for $K_{j}$ , the $L^{2}$ bound for $R_{j}$ and the off-support representation for compactly supported $L^{2}$ inputs; by steps 1.1 and 2.1 the kernel is standard $1$ -Hölder. Therefore $R_{j}$ is a standard-kernel Calderón–Zygmund operator in the sense of the base definition and the standard-kernel definition.

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Sources

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