Marcinkiewicz–Zygmund inequalities in quasi-Banach function spaces

Abstract

We obtain Marcinkiewicz-Zygmund (MZ) inequalities in various Banach and quasi-Banach spaces under minimal structural assumptions. 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 unified approach to deriving MZ inequalities not only for polynomials, but also for other function classes, including entire functions of exponential type, splines, exponential sums, and more. As applications, we derive error estimates for sampling operators, Nikolskii-type inequalities, as well as inequalities for best approximations and moduli of smoothness.