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.

翻译：我们在各类Banach空间与拟Banach空间中，以关于这些空间结构性质的最小假设为前提，获得了Marcinkiewicz-Zygmund（MZ）不等式。我们的主要结果表明，在一般拟Banach函数格$X$中的Bernstein不等式蕴含着$X$中的Marcinkiewicz-Zygmund型估计。我们提出了一种通用方法，不仅可针对多项式，还可针对其他函数类（包括指数型整函数、样条函数、指数和等）获得MZ不等式。