CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07‡ rests on unproved material

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Almost-everywhere convergence from the Carleson–Hunt estimate

Statement

Assume countable choice and let $1 < p < \infty$ . Suppose explicitly that a finite $C_{p} \geq 0$ satisfies $∥ C g ∥_{p} \leq C_{p} ∥ g ∥_{p}$ for every $g \in L^{p} (T)$ , with symmetric sums and normalized Haar measure. Then every $f \in L^{p} (T)$ satisfies $S_{N} f \to f$ almost everywhere.

Facts & Assumptions

Given: Countable choice, $1 < p < \infty$ , and the strong maximal estimate in the statement for all $g \in L^{p} (T)$ .

[F1]

Assuming countable choice, for fixed $1 \leq p < \infty$ , a bound $m {C g > λ} \leq A λ^{- p} ∥ g ∥_{p}^{p}$ for all $g \in L^{p}$ and all $λ > 0$ , with finite $A \geq 0$ , implies $S_{N} f \to f$ almost everywhere for every $f \in L^{p}$ (A weak maximal bound implies almost-everywhere Fourier convergence).

[F2]

If $h$ is nonnegative and measurable and $t > 0$ , then $m {h \geq t} \leq t^{- 1} \int h d m$ (Chebyshev-Markov inequality for the integral).

Proof

technique · strong-to-weak estimate and the maximal convergence principle

1.1

For $g \in L^{p}$ and $λ > 0$ , apply the integral inequality to the nonnegative measurable function $(C g)^{p}$ at $t = λ^{p}$ . The assumed norm bound makes its integral finite and yields $m {C g > λ} \leq λ^{- p} \int (C g)^{p} d m \leq C_{p}^{p} λ^{- p} ∥ g ∥_{p}^{p}$ .

F2given

2.1

Thus the weak estimate holds for every $g$ and every positive threshold with finite $A = C_{p}^{p} \geq 0$ . The fixed exponent is within $1 \leq p < \infty$ and countable choice is given, so the maximal convergence principle proves the assertion for every $f$ .

F1step 1.1given∎

Literature boundary

Carleson–Hunt maximal bound and almost-everywhere convergence — recorded theorem ‡ records that the strong estimate is true. This proof establishes the implication from that estimate as an explicit hypothesis; it does not prove or discharge the estimate itself.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

Laugesen, Harmonic Analysis Lecture Notes (standard reference, not scraped)