On L^pintegrability and convergence of trigonometric series

Abstract

We first give a necessary and sufficient condition for $x^{-\gamma }\phi ( x) \in L^{p}$, $1< p< \infty $, $1/p-1< \gamma < 1/p,$ where $\phi ( x) $ is the sum of either $\sum_{k=1}^{\infty }a_{k}\cos kx$ or $\sum_{k=1}^{\infty }b_{k}\sin kx$, under the