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arXiv · 2606.12920

An Oskolkov--Zhizhiashvili Criterion for Rectangular Fourier Sums

Ushangi Goginava
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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's classical sufficient conditions in the endpoint cases $p=1$ and $p=2$.

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Ushangi Goginava. 2026-06-11. An Oskolkov--Zhizhiashvili Criterion for Rectangular Fourier Sums. https://arxiv.org/abs/2606.12920

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