On the convergence of Fourier series (Q5902819)

We define the space \[ B^ p=\{f:(-\pi,\pi]\to R,\quad f(t)=\sum^{\infty}_{n=0}c_ nb_ n(t),\quad \sum^{\infty}_{n=0}| c_ n| <\infty \}. \] Each \(b_ n\) is a special p-atom, that is, a real valued function, defined on (- \(\pi\),\(\pi\) ], which is either \(b(t)=1/2\pi\) or \(b(t)=-1/| I|^{1/p}\chi_ R(t)+1/| I|^{1/p}\chi_ L(t),\) where I is an interval in (-\(\pi\),\(\pi\) ], L is the left half of I and R is the right half. \(| I|\) denotes the length of I and \(\chi_ E\) the characteristic function of E. \(B^ p\) is endowed with the norm \(\| f\|_{B^ p}=Int\sum^{\infty}_{n=0}| c_ n|\), where the infimum is taken over all possible representations of f. \(B^ p\) is a Banach space for \(1/2<p<\infty\). \(B^ p\) is continuously contained in \(L^ p\) for \(1\leq p<\infty\), but different. We have: Theorem. Let \(1<p<\infty\). If \(f\in B^ p\) then the maximal operator \(Tf(x)=\sup_{n}| S_ n(f,x)|\) maps \(B^ p\) into the Lorentz space L(p,1) boundedly, where \(S_ n(f,x)\) is the \(n^{th}\)-sum of the Fourier series of f.