Riesz means of Fourier series and integrals: Strong summability at the critical index

TL;DR

This paper investigates the strong summability of spherical Riesz means of Fourier series at critical indices, establishing sharp positive results for functions in various function spaces and extending to maximal operators on Hardy spaces.

Contribution

It provides new sharp results on strong summability of Riesz means at critical indices, including endpoint bounds on maximal operators for Hardy spaces.

Findings

Almost everywhere convergence may fail at the critical index for certain functions.

Established almost sharp theorems on strong summability for functions in L^p spaces.

Derived endpoint bounds for maximal operators on Hardy spaces.

Abstract

We consider spherical Riesz means of multiple Fourier series and some generalizations. While almost everywhere convergence of Riesz means at the critical index $(d-1)/2$ may fail for functions in the Hardy space $h^1(\mathbb T^d)$, we prove sharp positive results for strong summability almost everywhere. For functions in $L^p(\mathbb T^d)$, $1<p<2$, we consider Riesz means at the critical index $d(1/p-1/2)-1/2$ and prove an almost sharp theorem on strong summability. The results follow via transference from corresponding results for Fourier integrals. We include an endpoint bound on maximal operators associated with generalized Riesz means on Hardy spaces $H^p(\mathbb R^d)$ for $0<p<1$.