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From weak type weighted inequality to pointwise estimate for the decreasing rearrangement

Elona Agora, Jorge Antezana, Sergi Baena-Miret, María J. Carro

Abstract

We shall prove pointwise estimates for the decreasing rearrangement of $Tf$, where $T$ covers a wide range of interesting operators in Harmonic Analysis such as operators satisfying a Fefferman-Stein inequality, the Bochner-Riesz operator, rough operators, sparse operators, Fourier multipliers, etc. In particular, our main estimate is of the form $$ (Tf)^*(t) \leq C\left( \frac 1t\int_0^tf^*(s)\,ds + \int_t^\infty \left( 1 + \log\frac st \right)^{- 1}\varphi \left(1 + \log\frac st \right) f^*(s)\,\frac{ds}s \right), $$ where $\varphi$ is determined by the Muckenhoupt $A_p$-weight norm behaviour of the operator.

From weak type weighted inequality to pointwise estimate for the decreasing rearrangement

Abstract

We shall prove pointwise estimates for the decreasing rearrangement of , where covers a wide range of interesting operators in Harmonic Analysis such as operators satisfying a Fefferman-Stein inequality, the Bochner-Riesz operator, rough operators, sparse operators, Fourier multipliers, etc. In particular, our main estimate is of the form where is determined by the Muckenhoupt -weight norm behaviour of the operator.
Paper Structure (11 sections, 26 theorems, 149 equations)

This paper contains 11 sections, 26 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Definitions, previous results and lemmas
  3. Admissible functions
  4. Calderón type operators
  5. Lorentz spaces and some classes of weights
  6. Proof of our main results
  7. Examples and applications
  8. Fefferman-Stein inequality
  9. Bochner-Riesz and Rough singular operators
  10. Sparse Operators
  11. Fourier multipliers

Key Result

Theorem 1.1

Let $T$ be a sublinear operator such that, for every $u\in A_1$ and some $k \in \mathbb{N}$, $T$ satisfies bound with constant less than or equal to $C||u||_{A_1}^k$. Then, for every $t>0$ and every measurable function $f$,

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4
  • proof
  • Remark 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Theorem 1.9
  • ...and 33 more