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Marcinkiewicz-Zygmund inequalities in quasi-Banach function spaces

Yurii Kolomoitsev, Sergey Tikhonov

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.

Marcinkiewicz-Zygmund inequalities in quasi-Banach function spaces

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 implies Marcinkiewicz-Zygmund type estimates in . 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.

Paper Structure

This paper contains 29 sections, 42 theorems, 230 equations.

Table of Contents

  1. Introduction
  2. MZ-type inequalities in Banach and quasi-Banach spaces
  3. Boundedness of maximal and minimal functions
  4. General MZ-type inequalities in quasi-Banach spaces
  5. MZ-type inequalities with minimal number of nodes
  6. MZ-type inequalities under quadrature formulas and Bernstein's inequalities
  7. Approach to MZ inequalities using generalized translation
  8. MZ inequalities for trigonometric polynomials
  9. Minimal set of nodes
  10. MZ type inequalities in Banach lattices
  11. MZ type inequalities in quasi-Banach lattices
  12. More Marcinkiewicz--Zygmund inequalities in Orlicz spaces
  13. Historical notes
  14. MZ inequalities for algebraic polynomials
  15. MZ inequalities via Markov's inequality
  16. ...and 14 more sections

Key Result

Theorem 2.1

Let $X$ be a translation invariant Banach lattice. Assume that there exist ${\beta}>0$ and $B>0$ such that Then the following assertions hold: (A) For any $F_n\in \mathcal{F}_n$ and any $A>0$ satisfying we have (B) The left-hand side inequality in mmn holds for any $A>0$, that is, we do not require that $\eta<1$.

Theorems & Definitions (86)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6
  • proof
  • Remark 2.7
  • Remark 2.8
  • ...and 76 more