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Standard Hölder kernels satisfy the Hörmander condition

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Every standard $δ$ -Hölder Calderón–Zygmund kernel with constant $A_{2}^{'}$ (Standard (Hölder) Calderón–Zygmund kernels) is a Calderón–Zygmund kernel in the sense of the base definition (Calderón–Zygmund kernels and their associated operators), and its Hörmander constant may be taken to be $A_{2} = ∣ S^{n - 1} ∣ 2^{- δ} δ^{- 1} A_{2}^{'} .$

Facts & Assumptions

Given: Countable Choice; a standard $δ$ -Hölder Calderón–Zygmund kernel $k$ with constant $A_{2}^{'}$ , $0 < δ \leq 1$ , and its a priori annular constant $A_{1}$ ; a vector $y \neq 0$ .

[F1]

$k$ is measurable on $R^{n} ∖ {0}$ , integrable on compact subsets of $R^{n} ∖ {0}$ , satisfies the annular bound with $A_{1}$ , and satisfies $∣ k (x - y) - k (x) ∣ \leq A_{2}^{'} ∣ y ∣^{δ} ∣ x ∣^{- n - δ}$ whenever $∣ x ∣ \geq 2 ∣ y ∣ > 0$ (Standard (Hölder) Calderón–Zygmund kernels, Calderón–Zygmund kernels and their associated operators).

[F2]

Polar coordinates: for a Borel function $g \geq 0$ on $R^{n}$ , $\int_{R^{n}} g d λ = \int_{0}^{\infty} \int_{S^{n - 1}} g (r ω) d σ (ω) r^{n - 1} d r$ , where $σ$ is the surface measure with $σ (S^{n - 1}) = ∣ S^{n - 1} ∣$ (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma); the integral over a smaller domain is at most the integral over a larger one (Measures are monotone).

Proof

technique · direct

1.1F1F2algebra

Fix $y \neq 0$ . By the pointwise Hölder bound of [F1] and monotonicity of the integral, $\int_{∣ x ∣ \geq 2 ∣ y ∣} ∣ k (x - y) - k (x) ∣ d x \leq A_{2}^{'} ∣ y ∣^{δ} \int_{∣ x ∣ \geq 2 ∣ y ∣} ∣ x ∣^{- n - δ} d x .$

1.2F2algebra

Polar coordinates evaluate the radial integral: substituting $x = r ω$ and using $∣ x ∣^{- n - δ} r^{n - 1} = r^{- 1 - δ}$ , $\int_{∣ x ∣ \geq 2 ∣ y ∣} ∣ x ∣^{- n - δ} d x = ∣ S^{n - 1} ∣ \int_{2 ∣ y ∣}^{\infty} r^{- 1 - δ} d r = ∣ S^{n - 1} ∣ \frac{(2 ∣ y ∣)^{- δ}}{δ},$ the last step being the elementary integral $\int_{a}^{\infty} r^{- 1 - δ} d r = a^{- δ} / δ$ for $a > 0$ and $δ > 0$ .

2.1F1step 1.1step 1.2algebra∎

Combining steps 1.1 and 1.2 gives $\int_{∣ x ∣ \geq 2 ∣ y ∣} ∣ k (x - y) - k (x) ∣ d x \leq A_{2}^{'} ∣ y ∣^{δ} ∣ S^{n - 1} ∣ (2 ∣ y ∣)^{- δ} δ^{- 1} = ∣ S^{n - 1} ∣ 2^{- δ} δ^{- 1} A_{2}^{'}$ ; taking the supremum over $y \neq 0$ shows that Hörmander's condition holds with $A_{2} = ∣ S^{n - 1} ∣ 2^{- δ} δ^{- 1} A_{2}^{'}$ , while the annular bound holds with $A_{1}$ by hypothesis. Hence $k$ is a Calderón–Zygmund kernel in the base sense with the stated Hörmander constant.

Depends on

Used by

The Hilbert transform is bounded on Lp Corollary
The Riesz transforms are bounded on Lp Corollary
Maximal truncations: weak (1,1) and strong Lp bounds Theorem

Dependency tree · two levels

23 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)
Mark Williams, Notes on Harmonic Analysis (standard reference, not scraped)