Carleson maximal partial-sum operator

Assume the Axiom of Countable Choice for the torus $L^1$ integrals. Use $\mathbb{T}=\mathbb{R}\mathrm{/}\mathbb{Z}$ with Haar measure $m(\mathbb{T})=1$ and the negative-sign Fourier coefficients and symmetric partial sums of Period-one Fourier coefficients, partial sums, and convolution on the torus. For $f\in L^1(\mathbb{T})$ define the Carleson maximal partial-sum operator by

$Cf(x):={\sup }_{N\in {\mathbb{Z}}_{\ge 0}}\mathrm{\mid }{S}_{N}f(x)\mathrm{\mid }\in [0,\mathrm{\infty }]\mathrm{.}$

Each coefficient is a finite integral independent of the representative of $f$, and each $S_{N}f$ is a continuous trigonometric polynomial. Thus $C$ depends only on the $L^1$ class, at every point. For every real $a$, the set $\{Cf>a\}={\bigcup }_{N\ge 0}\{\mathrm{\mid }{S}_{N}f\mathrm{\mid }>a\}$ is measurable.

Linearity of finite Fourier sums gives $C(f+g)\le Cf+Cg$ and $C(af)=\mathrm{\mid }a\mathrm{\mid }Cf$ for nonzero scalars $a$. Also $C0=0$; with $0\cdot \mathrm{\infty }:=0$ the homogeneity identity holds for $a=0$ as well. This is sublinearity with extended nonnegative values. Including $N=0$ retains the constant Fourier coefficient.