An Oskolkov--Zhizhiashvili Criterion for Rectangular Fourier Sums

Abstract

Let $S_{\mathbf n}f$ denote the symmetric rectangular partial sums of the trigonometric Fourier series of a function on the $d$-dimensional torus. We prove a summable endpoint criterion at the Zhizhiashvili critical scale for all $d\ge2$ and $1\le p\le2$. The criterion allows a general summable secondary weight at the iterated-logarithmic level and contains, as special cases, a double-logarithmic endpoint criterion and an $L^p$ Oskolkov-type corollary. In particular, it answers the Zhizhiashvili--Marcinkiewicz problem for $1<p<2$ and sharpens Zhizhiashvili&#39;s classical sufficient conditions in the endpoint cases $p=1$ and $p=2$.