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The Hilbert and Riesz transforms are Calderon-Zygmund operators

Statement

Assume Countable Choice. Let $H$ be the Hilbert transform of the line and, for $1 \leq j \leq n$ , let $R_{j}$ be the Riesz transform of Riesz transforms on Euclidean space. Then $H$ and each $R_{j}$ are Calderon-Zygmund operators with the kernels $k (x) = 1 / (π x)$ and $K_{j} (x) = c_{n} x_{j} / ∣ x ∣^{n + 1}$ : they are $L^{2}$ -bounded with norm at most $1$ , and for every compactly supported $f \in L^{2}$ and almost every $x \notin supp f$ one has $T f (x) = \int k (x - y) f (y) d y$ with the kernel of $T$ . Moreover both kernels are standard $1$ -Holder Calderon-Zygmund kernels with the published constants ( $A_{2}^{'} = 2 / π$ and $C_{n} = c_{n} 2^{n + 1} (3 n + 4)$ , respectively).

Facts & Assumptions

Given: Countable Choice, the Hilbert transform $H$ on $R$ with kernel $k (x) = 1 / (π x)$ , and the Riesz transforms $R_{1}, \dots, R_{n}$ on $R^{n}$ with kernels $K_{j} (x) = c_{n} x_{j} / ∣ x ∣^{n + 1}$ .

[F1]

The Hilbert transform is a well-defined $L^{2}$ operator with $∥ H f ∥_{2} = ∥ f ∥_{2}$ and $H^{2} f = - f$ ; it is skew-adjoint, $⟨ H f, g ⟩ = - ⟨ f, H g ⟩$ ; on Schwartz functions $H f = W * f$ , the principal value $\lim_{ε ↓ 0} H_{ε} f (x)$ exists at every $x$ and equals $(W * f) (x)$ , with $F (H f) = - i sgn (ξ) \hat{f} (ξ)$ (The Hilbert transform is an L2 isometry and squares to minus the identity, The Hilbert transform is skew-adjoint on L2, The Hilbert transform is the tempered convolution with pv(1/(pi x)) and has signum Fourier multiplier, Truncated Hilbert transform and principal value).

[F2]

The Riesz transform $R_{j}$ has multiplier $m_{j} (ξ) = - i ξ_{j} / ∣ ξ ∣$ for $ξ \neq 0$ (and $m_{j} (0) = 0$ ), acts on Schwartz functions, and satisfies $∥ R_{j} f ∥_{2} \leq ∥ f ∥_{2}$ for all $f \in L^{2}$ ; its kernel $K_{j} (x) = c_{n} x_{j} / ∣ x ∣^{n + 1}$ obeys $∣ K_{j} (x) ∣ \leq c_{n} ∣ x ∣^{- n}$ and $∣ K_{j} (x - h) - K_{j} (x) ∣ \leq C_{n} ∣ h ∣ ∣ x ∣^{- (n + 1)}$ whenever $x \neq 0$ and $∣ h ∣ \leq ∣ x ∣ / 2$ , with $C_{n} = c_{n} 2^{n + 1} (3 n + 4)$ ; and for every Schwartz function $f$ the truncated integrals $\int_{∣ y ∣ > ε} K_{j} (y) f (x - y) d y$ converge as $ε ↓ 0$ , for every $x$ , to a continuous representative of the $L^{2}$ class $R_{j} f$ (Riesz transforms on Euclidean space, Riesz transforms are L2 contractions and square to minus the identity in sum, Riesz kernel size, difference and spherical-cancellation bounds, The Riesz transform is the principal value of its kernel, with the matching constant).

[F3]

Polar coordinates under Countable Choice: for every Borel measurable nonnegative $g$ on $R^{n}$ , $\int_{R^{n}} g (x) d x = \int_{0}^{\infty} \int_{S^{n - 1}} g (r ω) r^{n - 1} d σ (ω) d r,$ where $σ (S^{n - 1}) = ∣ S^{n - 1} ∣$ (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma).

[F4]

The Fourier transform extends to a unitary $F_{2} : L^{2} (R^{n}) \to L^{2} (R^{n})$ preserving the first-variable-linear inner product (Plancherel theorem).

[F5]

The Fubini theorem applies to $L^{1}$ functions on products of $σ$ -finite measure spaces (Fubini's theorem for L^1 functions on a sigma-finite product).

[F6]

The map sending a locally integrable function to its regular distribution is injective on $L_{l o c}^{1} (Ω)$ modulo almost-everywhere equality, for every open $Ω$ (Locally integrable functions embed in distributions).

[F7]

The Calderon-Zygmund kernel and operator conventions are those of Calderón–Zygmund kernels and their associated operators: conditions (1) and (2) are the annular size and Hormander conditions, and condition (3) is the off-support representation by the kernel.

[F9]

The pointwise first-difference estimate defines the standard $1$ -Holder kernel condition once the base Calderon-Zygmund conditions have been verified (Standard (Hölder) Calderón–Zygmund kernels).

[F8]

Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

1.1F1F7F8algebra

The Hilbert kernel $k (x) = 1 / (π x)$ is odd, continuous on $R ∖ {0}$ and hence locally integrable there, and for every $R > 0$ the annular integral is $\int_{R \leq ∣ x ∣ \leq 2 R} ∣ k (x) ∣ d x = \frac{1}{π} \int_{R \leq ∣ x ∣ \leq 2 R} \frac{d x}{∣ x ∣} = \frac{2 \ln 2}{π}$ , so condition (1) of [F7] holds with $A_{1} = 2 \ln 2 / π$ . If $∣ x ∣ \geq 2 ∣ h ∣ > 0$ then $∣ x - h ∣ \geq ∣ x ∣ / 2$ , so $∣ k (x - h) - k (x) ∣ = \frac{∣ h ∣}{π ∣ x ∣ ∣ x - h ∣} \leq \frac{2}{π} \frac{∣ h ∣}{∣ x ∣^{2}}$ ; this gives the required pointwise first-difference bound with constant $A_{2}^{'} = 2 / π$ . The standing Countable Choice hypothesis [F8] is inherited with the $L^{2}$ Hilbert theory of [F1], which is stated under it, and is used nowhere else in this step.

1.2F2F7F8algebra

The Riesz kernel $K_{j} (x) = c_{n} x_{j} / ∣ x ∣^{n + 1}$ is smooth and odd on $R^{n} ∖ {0}$ , hence locally integrable there, and $∣ K_{j} (x) ∣ \leq c_{n} ∣ x ∣^{- n}$ gives $\int_{R \leq ∣ x ∣ \leq 2 R} ∣ K_{j} (x) ∣ d x \leq c_{n} ∣ S^{n - 1} ∣ \ln 2$ for every $R > 0$ , so condition (1) of [F7] holds with $A_{1} = c_{n} ∣ S^{n - 1} ∣ \ln 2$ ; [F2] gives the pointwise first-difference bound with $A_{2}^{'} = C_{n} = c_{n} 2^{n + 1} (3 n + 4)$ . The standing Countable Choice hypothesis [F8] is inherited here with the Riesz theory of [F2], which is stated under it.

1.3F1F2F4algebra

The operators are $L^{2}$ -bounded with norm at most $1$ : $∥ H f ∥_{2} = ∥ f ∥_{2}$ and $∥ R_{j} f ∥_{2} \leq ∥ f ∥_{2}$ by [F1] and [F2]. The Hilbert transform is skew-adjoint by [F1]. For the Riesz transforms, Plancherel [F4] and the multiplier representation give, for $f, g \in L^{2} (R^{n})$ , $⟨ R_{j} f, g ⟩ = \int_{R^{n}} m_{j} \hat{f} \overline{\hat{g}} d ξ = \int_{R^{n}} \hat{f} \overline{(- m_{j}) \hat{g}} d ξ = ⟨ f, - R_{j} g ⟩$ , because $\overline{m_{j}} = - m_{j}$ on the purely imaginary symbol; hence $R_{j}^{*} = - R_{j}$ .

2.1F3step 1.1step 1.2F7F9algebra

Both kernels satisfy the annular size condition (1) of [F7] by steps 1.1 and 1.2. For either kernel $q$ and every $h \neq 0$ , the pointwise difference bound in step 1.1 or 1.2 gives $∣ q (x - h) - q (x) ∣ \leq A_{2}^{'} ∣ h ∣ ∣ x ∣^{- n - 1}$ for $∣ x ∣ \geq 2 ∣ h ∣$ . By polar coordinates [F3], $\int_{∣ x ∣ \geq 2 ∣ h ∣} ∣ q (x - h) - q (x) ∣ d x \leq A_{2}^{'} ∣ h ∣ \int_{∣ x ∣ \geq 2 ∣ h ∣} ∣ x ∣^{- n - 1} d x = A_{2}^{'} ∣ h ∣ ∣ S^{n - 1} ∣ \int_{2 ∣ h ∣}^{\infty} r^{- 2} d r = \frac{∣ S^{n - 1} ∣}{2} A_{2}^{'} .$ Thus condition (2) of [F7] holds directly with $A_{2} = ∣ S^{n - 1} ∣ A_{2}^{'} / 2$ , so both kernels are Calderon-Zygmund kernels in the base sense. Since condition (2) is now established, their pointwise first-difference bounds make them standard $1$ -Holder kernels by Standard (Hölder) Calderón–Zygmund kernels, with constants $2 / π$ and $C_{n}$ .

2.2F1F2F5step 1.3algebra

Fix a compactly supported $f \in L^{2} (R^{n})$ and put $Ω = R^{n} ∖ supp f$ ; fix also $φ \in C_{c}^{\infty} (Ω)$ . Then $supp f$ and $supp φ$ are disjoint compact sets, so their distance is positive and the kernel is bounded on $supp f \times supp φ$ ; since $f$ is integrable on its compact support by Cauchy-Schwarz, the double integral below is absolutely convergent. For $x \in supp f$ one has $x \notin supp φ$ , and the truncated integrals in the principal-value formulas [F1] and [F2] converge to the full absolutely convergent integral $\int k (x - y) φ (y) d y$ , so $T φ (x) = \int k (x - y) φ (y) d y$ there. Skew-adjointness from step 1.3 therefore gives $⟨ T f, φ ⟩ = - ⟨ f, T φ ⟩ = - \int_{supp f} f (x) \overline{\int k (x - y) φ (y) d y} d x$ , and since the kernels are real-valued this equals $- \iint f (x) k (x - y) \overline{φ (y)} d y d x$ . Substituting $k (x - y) = - k (y - x)$ by oddness and applying Fubini [F5] yields $⟨ T f, φ ⟩ = \int \overline{φ (y)} g (y) d y$ with $g (y) : = \int k (y - x) f (x) d x$ .

3.1step 2.2F6

For every $y \in Ω$ the integral defining $g (y)$ is absolutely convergent: $x \mapsto k (y - x)$ is bounded on the compact set $supp f$ , and $\int_{supp f} ∣ f (x) ∣ d x \leq ∣ supp f ∣^{1 / 2} ∥ f ∥_{2} < \infty$ . Hence $g$ is locally integrable on the open set $Ω$ , and $T f ∣_{Ω}$ is locally integrable as well because $T f \in L^{2} (R^{n})$ ; step 2.2 says that the regular distributions of $T f ∣_{Ω}$ and $g ∣_{Ω}$ agree on every test function supported in $Ω$ , so the injectivity of [F6], applied on $Ω$ , gives $T f = g$ almost everywhere on $Ω$ .

4.1step 2.1step 1.3step 3.1F7∎

By step 2.1 the kernels $k$ and $K_{j}$ are Calderon-Zygmund kernels, by step 1.3 the operators $H$ and $R_{j}$ are $L^{2}$ -bounded with norm at most $1$ and skew-adjoint, and by step 3.1 the off-support representation (3) of [F7] holds: for almost every $x \notin supp f$ , $T f (x) = \int k (x - y) f (y) d y$ . Therefore $H$ and $R_{j}$ are Calderon-Zygmund operators with the kernels $k$ and $K_{j}$ , and the standard $1$ -Holder constants are $A_{2}^{'} = 2 / π$ and $C_{n} = c_{n} 2^{n + 1} (3 n + 4)$ .

Depends on

Used by

The Hilbert and Riesz transforms map L-infinity to BMO 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

Mark Williams, Notes on Harmonic Analysis (January 11, 2022) (standard reference, not scraped)
Terence Tao, Math 247A Lecture Notes 4 (UCLA, Fall 2006) (standard reference, not scraped)