Marcinkiewicz-Zygmund inequalities in quasi-Banach function spaces

TL;DR

This paper establishes Marcinkiewicz-Zygmund inequalities in a broad class of Banach and quasi-Banach spaces, linking Bernstein inequalities to MZ estimates and extending results to various function classes.

Contribution

It introduces a general framework connecting Bernstein and MZ inequalities in quasi-Banach function spaces, applicable to multiple function classes.

Findings

MZ inequalities hold under minimal structural assumptions.

Bernstein inequality implies MZ estimates in quasi-Banach spaces.

Applicable to polynomials, entire functions, splines, and exponential sums.

Abstract

We obtain Marcinkiewicz--ygmund (MZ) inequalities in various Banach and quasi-Banach spaces under minimal assumptions on the structural properties of these spaces. Our main results show that the Bernstein inequality in a general quasi-Banach function lattice $X$ implies Marcinkiewicz-Zygmund type estimates in $X$. We present a general approach to obtain MZ inequalities not only for polynomials but for other function classes including entire functions of exponential type, splines, exponential sums, etc.