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Almost-everywhere convergence of principal-value truncations

Statement

Assume Countable Choice. Fix $1 \leq p < \infty$ and let $k$ and $T$ be as in Maximal truncations: weak (1,1) and strong Lp bounds: $k$ satisfies the pointwise size bound with constant $A_{1}$ , the standard $δ$ -Hölder bound with constant $A_{2}^{'}$ and the cancellation bound $A_{3}$ , and $T$ is the associated $L^{2}$ -bounded operator with off-support representation and $L^{2}$ norm $B$ . Let $D \subseteq L^{p} (R^{n}; C)$ be dense in $L^{p} (R^{n}; C)$ and suppose that for every $g \in D$ the limit $\lim_{ε ↓ 0} T_{ε} g (x)$ exists for almost every $x \in R^{n}$ . Then for every $f \in L^{p} (R^{n}; C)$ the limit $\lim_{ε ↓ 0} T_{ε} f (x)$ exists for almost every $x \in R^{n}$ .

In particular, for the Hilbert kernel $1 / (π x)$ on $R$ and the Riesz kernels $c_{n} x_{j} / ∣ x ∣^{n + 1}$ on $R^{n}$ the dense class $D = S (R^{n})$ satisfies the hypothesis, by the published principal-value formulas for Schwartz functions.

Facts & Assumptions

Given: Countable Choice; $1 \leq p < \infty$ ; $k$ , $T$ as in the statement; a dense subspace $D \subseteq L^{p}$ such that $\lim_{ε ↓ 0} T_{ε} g (x)$ exists a.e. for every $g \in D$ ; a function $f \in L^{p}$ and $λ > 0$ .

[F1]

For $f \in L^{p}$ the truncations $T_{ε} f$ , $ε > 0$ , are defined pointwise by absolutely convergent integrals and $T^{*} f = \sup_{ε > 0} ∣ T_{ε} f ∣$ (Maximal truncated singular integrals); $T^{*} f \leq T^{* *} f$ , and the maximal truncation $T^{* *}$ satisfies $∣ {∣ T^{* *} h ∣ > λ} ∣ \leq C_{n, δ} (A_{1} + A_{2}^{'} + A_{3} + B) λ^{- 1} ∥ h ∥_{1}$ for $h \in L^{1}$ and $∥ T^{* *} h ∥_{p} \leq C_{n, p, δ} (A_{1} + A_{2}^{'} + A_{3} + B) \max (p, (p - 1)^{- 1}) ∥ h ∥_{p}$ for $1 < p < \infty$ , hence the same bounds hold for $T^{*}$ (Maximal truncations: weak (1,1) and strong Lp bounds).

[F2]

Chebyshev's inequality: for measurable $u$ and $t > 0$ , $∣ {∣ u ∣ > t} ∣ \leq t^{- p} \int ∣ u ∣^{p}$ when $u \in L^{p}$ (Chebyshev-Markov inequality for the integral); $C_{c}^{\infty} (R^{n})$ — and hence its superset $S (R^{n})$ — is dense in $L^{p} (R^{n}; C)$ for $1 \leq p < \infty$ (Complex finite-simple and smooth compact-support density for finite p); the $L^{p}$ conventions are those of the maximal-truncation theorem and Countable Choice is The Axiom of Countable Choice ( ${A C}_{ω}$ ).

[F3]

For the Hilbert and Riesz kernels the principal-value truncations converge on every Schwartz input and identify the $L^{2}$ operators. Their kernel size, first-difference, cancellation, $L^{2}$ and off-support operator conditions are proved in the corresponding items. (The Hilbert transform is the tempered convolution with pv(1/(pi x)) and has signum Fourier multiplier, The Riesz transform is the principal value of its kernel, with the matching constant, Truncated Hilbert transform and principal value, Riesz transforms on Euclidean space, The Hilbert transform is bounded on Lp, The Riesz transforms are bounded on Lp, Riesz kernel size, difference and spherical-cancellation bounds)

Proof

technique · direct

1.1F1givenalgebra

Oscillation bound. Put $H_{f} (x) : = {lim sup}_{ε, θ ↓ 0} ∣ T_{ε} f (x) - T_{θ} f (x) ∣$ . For every $g \in D$ , on the full-measure set where $(T_{ε} g (x))_{ε}$ converges, the triangle inequality gives $∣ T_{ε} f - T_{θ} f ∣ \leq ∣ T_{ε} (f - g) ∣ + ∣ T_{θ} (f - g) ∣ + ∣ T_{ε} g - T_{θ} g ∣ \leq 2 T^{*} (f - g) (x) + ∣ T_{ε} g - T_{θ} g ∣$ , and the last term tends to $0$ as $ε, θ ↓ 0$ ; hence $H_{f} \leq 2 T^{*} (f - g)$ almost everywhere.

2.1F1step 1.1algebra

The case $p = 1$ . Taking $g \in D$ and using step 1.1 and the weak $(1, 1)$ bound of [F1] for $f - g \in L^{1}$ , $∣ {H_{f} > λ} ∣ \leq ∣ {T^{*} (f - g) > λ / 2} ∣ \leq 2 C_{n, δ} (A_{1} + A_{2}^{'} + A_{3} + B) λ^{- 1} ∥ f - g ∥_{1}$ for every $λ > 0$ ; since $D$ is dense in $L^{1}$ , the infimum over $g \in D$ gives $∣ {H_{f} > λ} ∣ = 0$ for every $λ > 0$ , hence $H_{f} = 0$ almost everywhere. Thus $(T_{ε} f (x))_{ε > 0}$ is a Cauchy family as $ε ↓ 0$ for almost every $x$ , so its limit exists almost everywhere.

2.2F1F2step 1.1algebra

The case $1 < p < \infty$ . With $g \in D$ , step 1.1, Chebyshev's inequality and the strong $L^{p}$ bound of [F1] give $∣ {H_{f} > λ} ∣ \leq ∣ {T^{*} (f - g) > λ / 2} ∣ \leq (2 / λ)^{p} ∥ T^{*} (f - g) ∥_{p}^{p} \leq (2 C_{n, p, δ} (A_{1} + A_{2}^{'} + A_{3} + B) \max (p, (p - 1)^{- 1}) / λ)^{p} ∥ f - g ∥_{p}^{p},$ and letting $g \to f$ in $L^{p}$ through $D$ gives $∣ {H_{f} > λ} ∣ = 0$ for every $λ > 0$ , hence $H_{f} = 0$ almost everywhere and the limit exists almost everywhere.

3.1F2F3step 2.1step 2.2∎

The Hilbert and Riesz kernels have the size, Hölder, spherical cancellation, $L^{2}$ bound and off-support representation in [F3]. Zero spherical means give the annular cancellation bound $A_{3} = 0$ . Their principal-value distributions are defined by subtracting a test's value at zero on $∣ y ∣ < 1$ ; the size estimate makes the resulting integrand integrable, bounded by $C ∣ y ∣^{1 - n}$ there, and Schwartz decay controls infinity. Thus they satisfy the maximal theorem's hypotheses. The dense class $S$ has convergence at every point by [F3], and is dense in each finite-exponent $L^{p}$ by [F2]. Steps 2.1 and 2.2 therefore give the asserted almost-everywhere convergence for every $f \in L^{p}$ .

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Sources

Mark Williams, Notes on Harmonic Analysis (standard reference, not scraped)
Loukas Grafakos, Classical Fourier Analysis, third edition (standard reference, not scraped)