An Oskolkov--Zhizhiashvili Criterion for Rectangular Fourier Sums

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 $d\ge 2$ and $1\le p\le 2$. 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’s classical sufficient conditions in the endpoint cases $p=1$ and $p=2$.