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Vector-valued estimates for shifted operators

Bae Jun Park

Abstract

Shifted variants of (dyadic) Hardy-Littlewood maximal function and Stein's square function have played a significant role in the study of many important operators such as Calderon commutators, (bilinear) Hilbert transforms, multilinear multipliers, and multilinear rough singular integrals. Estimates for such shifted operators have a certain logarithmic growth in terms of the shift factor, but the optimality of the logarithmic growth has not yet been fully resolved. In this article, we provide sharp vector-valued shifted maximal inequality for generalized Peetre's maximal function, from which improved estimates for the above shifted operators follow with optimal logarithmic growths in a new way. We also obtain a vector-valued maximal inequality for the shifted (dyadic) Hardy-Littlewood maximal operator.

Vector-valued estimates for shifted operators

Abstract

Shifted variants of (dyadic) Hardy-Littlewood maximal function and Stein's square function have played a significant role in the study of many important operators such as Calderon commutators, (bilinear) Hilbert transforms, multilinear multipliers, and multilinear rough singular integrals. Estimates for such shifted operators have a certain logarithmic growth in terms of the shift factor, but the optimality of the logarithmic growth has not yet been fully resolved. In this article, we provide sharp vector-valued shifted maximal inequality for generalized Peetre's maximal function, from which improved estimates for the above shifted operators follow with optimal logarithmic growths in a new way. We also obtain a vector-valued maximal inequality for the shifted (dyadic) Hardy-Littlewood maximal operator.
Paper Structure (29 sections, 10 theorems, 205 equations)

This paper contains 29 sections, 10 theorems, 205 equations.

Table of Contents

  1. Introduction
  2. Shifted dyadic maximal function estimates
  3. Shifted function estimates for Hardy spaces and $BMO$
  4. Proof of Proposition \ref{['maintheorem']}
  5. The case when $0<q<p< \infty$
  6. Proof of \ref{['qpclaim1']}
  7. Proof of \ref{['qpclaim2']}
  8. The case when $0<p<q\le \infty$
  9. The case when $p=\infty$ and $0<q< \infty$
  10. Proof of Theorem \ref{['dyadmaxtheorem']}
  11. Key lemmas for the proof of Theorem \ref{['vectorvalueddyadic']}
  12. Vector-valued extension of boundedness for positive linearizable operators
  13. Calderón-Zygmund theory for vector-valued function spaces
  14. Proof of Theorem \ref{['vectorvalueddyadic']}
  15. The case when $p\le q\le \infty$
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $0<p,q,t\le \infty$, $y\in\mathbb R^n$, and $\sigma> n/\min{(p,q,t)}$. For each $k\in\mathbb{N}$, let $f_k\in \mathcal{E}(A2^k)$ for some $A>0$.

Theorems & Definitions (13)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Lemma 4.1
  • Lemma 4.2
  • ...and 3 more