On the convergence of Fourier series

Abstract

We define the space Bp={f:(−π,π]→R, f(t)=∑n=0∞cnbn(t), ∑n=0∞|cn|<∞}. Each bn is a special p-atom, that is, a real valued function, defined on (−π,π], which is either b(t)=1/2π or b(t)=−1|I|1/pXR(t)+1|I|1/pXL(t), where I is an interval in (−π,π], L is the left half of I and R is the right half. |I| denotes the length of I and XE the characteristic function of E. Bp is endowed with the norm ‖f‖Bp=Int∑n=0∞|cn|, where the infimum is taken over all possible representations of f. Bp is a Banach space for 1/2<p<∞. Bp is continuously contained in Lp for 1≤p<∞, but different. We have THEOREM. Let 1<p<∞. If f∈Bp then the maximal operator Tf(x)=supn|Sn(f,x)| maps Bp into the Lorentz space L(p,1) boundedly, where Sn(f,x) is the nth-sum of the Fourier Series of f.

Keywords

• maximal operator
• Lorentz spaces and Fourier series.