Vector-valued inequalities for Fourier series

Abstract

Denoting by S ∗ {S^\ast } the maximal partial sum operator of Fourier series, we prove that S ∗ ( f 1 , f 2 , … , f k , … ) = ( S ∗ f 1 , S ∗ f 2 , … , S ∗ f k , … ) {S^\ast }({f_1},{f_2}, \ldots ,{f_k}, \ldots ) = ({S^\ast }{f_1},{S^\ast }{f_2}, \ldots ,{S^\ast }{f_k}, \ldots ) is a bounded operator from L p ( l r ) {L^p}({l^r}) to itself, 1 > p , r > ∞ 1 > p,r > \infty . Thus, we extend the theorem of Carleson and Hunt on pointwise convergence of Fourier series to the case of vector valued functions. We give also an application to the rectangular convergence of double Fourier series.