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Almost-everywhere convergence from the Carleson–Hunt estimate

Statement

Assume the Axiom of Choice and let $1 < p < \infty$ . Every complex $f \in L^{p} (T)$ satisfies $S_{N} f \to f$ almost everywhere, with symmetric partial sums and normalized Haar measure on the period-one torus. The conclusion holds for every measurable representative of $f$ .

Facts & Assumptions

Given: The Axiom of Choice, $1 < p < \infty$ , and the normalized torus conventions.

[F1]

Under AC, a finite constant $C_{p}$ satisfies $∥ C g ∥_{p} \leq C_{p} ∥ g ∥_{p}$ for every complex $g \in L^{p} (T)$ , where $C g = \sup_{N \geq 0} ∣ S_{N} g ∣$ (Carleson hunt maximal inequality on the torus).

[F2]

Under countable choice, 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 $f \in L^{p}$ , for every measurable representative (A weak maximal bound implies almost-everywhere Fourier convergence).

[F3]

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

[F4]

The Axiom of Choice (The Axiom of Choice) implies countable choice and supplies the choice hypothesis of F1 and F2.

Proof

1.1F1F3given

For $g \in L^{p}$ and $λ > 0$ , F1 and F3 applied to $(C g)^{p}$ at $t = λ^{p}$ give $m {C g > λ} \leq λ^{- p} \int (C g)^{p} d m \leq C_{p}^{p} λ^{- p} ∥ g ∥_{p}^{p}$ . The maximal function and Fourier conventions in F1 agree with those in F2.

2.1F2F4step 1.1given∎

The weak estimate therefore holds for every $g$ and every positive threshold with finite $A = C_{p}^{p}$ . Since $1 < p < \infty$ lies in the range $1 \leq p < \infty$ of F2 and AC supplies countable choice, F2 proves the assertion for every $f$ and every measurable representative. The strong estimate is discharged by the local proved theorem F1.

Depends on

Used by

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Sources

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