Carleson–Hunt maximal bound and almost-everywhere convergence — recorded theorem

On the period-one torus with normalized Haar measure, for every $1<p<\mathrm{\infty }$ there is a finite constant $C_p$ such that

$\mathrm{\parallel }Cf{\mathrm{\parallel }}_p\le C_p\mathrm{\parallel }f{\mathrm{\parallel }}_p(f\in L^p(\mathbb{T})),$

where $C$ is Carleson maximal partial-sum operator. Consequently,

$S_{N}f(x)⟶f(x)\text{for almost every }x\in \mathbb{T}\mathrm{.}$

Carleson's original result concerns $p=2$; Hunt obtained the full open range. This is recorded literature, with no local proof of the maximal estimate. Laugesen's Theorem 8.7 and its omitted-proof discussion give the torus convergence statement and the required strong maximal estimate; the source's period $2\pi$ is rescaled by $t=2\pi x$ with normalized measure.

The norm here is $\mathrm{\parallel }{\sup }_N\mathrm{\mid }{S}_{N}f\mathrm{\mid }{\mathrm{\parallel }}_p$. Uniformly bounding the separate numbers $\mathrm{\parallel }{S}_{N}f{\mathrm{\parallel }}_p$ does not supply this estimate. The endpoint $p=1$ is excluded.