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A Christ-Kiselev maximal theorem in quasi-Banach function lattices

Mieczysław Mastyło, Gord Sinnamon

TL;DR

This work extends the Christ-Kiselev maximal theorem from classical L^p spaces to the broader framework of quasi-Banach function lattices by leveraging upper and lower p-estimates, as well as duality and interpolation tools. It provides two CK-type formulations: a quantitative two-term-estimate version with explicit constants and a qualitative version under general lower p-estimates and upper p-estimates, with constants independent of the filtration. The theory is developed for Lorentz spaces and Weiner amalgams, yielding maximal-operator bounds for the Fourier transform and related operators, and is extended to Köthe dual operators and interpolation spaces. Duality and interpolation are further exploited to transfer CK-type bounds to dual settings and to interpolated spaces, with applications to Lorentz Γ-spaces and convolution operators of Bak–Seeger type, broadening maximal inequality techniques in harmonic analysis. These results offer refined maximal inequalities across refined function spaces, enabling sharper Fourier-analytic estimates in quasi-Banach settings.

Abstract

A Christ-Kiselev maximal theorem is proved for linear operators between quasi-Banach function lattices satisfying certain lattice geometrical conditions. The result is further explored for weighted Lorentz spaces, classical Lorentz spaces, and Wiener amalgams of Lebesgue function and sequence spaces. Extensions are made to Köthe dual operators and to operators on interpolation spaces of quasi-Banach function lattices. Several applications to maximal Fourier operators are presented.

A Christ-Kiselev maximal theorem in quasi-Banach function lattices

TL;DR

This work extends the Christ-Kiselev maximal theorem from classical L^p spaces to the broader framework of quasi-Banach function lattices by leveraging upper and lower p-estimates, as well as duality and interpolation tools. It provides two CK-type formulations: a quantitative two-term-estimate version with explicit constants and a qualitative version under general lower p-estimates and upper p-estimates, with constants independent of the filtration. The theory is developed for Lorentz spaces and Weiner amalgams, yielding maximal-operator bounds for the Fourier transform and related operators, and is extended to Köthe dual operators and interpolation spaces. Duality and interpolation are further exploited to transfer CK-type bounds to dual settings and to interpolated spaces, with applications to Lorentz Γ-spaces and convolution operators of Bak–Seeger type, broadening maximal inequality techniques in harmonic analysis. These results offer refined maximal inequalities across refined function spaces, enabling sharper Fourier-analytic estimates in quasi-Banach settings.

Abstract

A Christ-Kiselev maximal theorem is proved for linear operators between quasi-Banach function lattices satisfying certain lattice geometrical conditions. The result is further explored for weighted Lorentz spaces, classical Lorentz spaces, and Wiener amalgams of Lebesgue function and sequence spaces. Extensions are made to Köthe dual operators and to operators on interpolation spaces of quasi-Banach function lattices. Several applications to maximal Fourier operators are presented.
Paper Structure (13 sections, 23 theorems, 102 equations)

This paper contains 13 sections, 23 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Upper and lower $p$-estimates in function lattices
  4. A Christ-Kiselev theorem in quasi-Banach lattices
  5. In Lorentz spaces
  6. Application: The Fourier Transform in Lorentz spaces
  7. Application: Consequences of a result of Sledd and Stegenga
  8. In Weiner amalgams
  9. Application: A Menshov-Paley-Zygmund theorem
  10. For Köthe dual operators
  11. Application: The Fourier Transform in Lorentz $\Gamma$-spaces
  12. Interpolation of Christ-Kiselev maximal operators
  13. Application: A convolution operator of Bak and Seeger

Key Result

Theorem 1.1

Let $(\Omega, \mu)$ and $(\widetilde{\Omega}, \nu)$ be measure spaces and let $\mathcal{A}$ be a countable, totally ordered collection of measurable subsets of $\Omega$. If $1\le p<q\le\infty$, and $T$ is a bounded, linear operator from $L^p_\mu$ to $L^q_\nu$ then the sublinear operator $T^*$, defin is also bounded from $L^p_\mu$ to $L^q_\nu$ and $\|T^{*}\| \le (1- 2^{1/q-1/p})^{-1}\|T\|.$

Theorems & Definitions (45)

  • Theorem 1.1: Christ-Kiselev Maximal Theorem
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 3.1
  • ...and 35 more