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Characterization of the weak-type boundedness of the Hilbert transform on weighted Lorentz spaces

Elona Agora, María J. Carro, Javier Soria

Abstract

We characterize the weak-type boundedness of the Hilbert transform $H$ on weighted Lorentz spaces $Λ^p_u(w)$, with $p>0$, in terms of some geometric conditions on the weights $u$ and $w$ and the weak-type boundedness of the Hardy-Littlewood maximal operator on the same spaces. Our results recover simultaneously the theory of the boundedness of $H$ on weighted Lebesgue spaces $L^p(u)$ and Muckenhoupt weights $A_p$, and the theory on classical Lorentz spaces $Λ^p(w)$ and Ariño Muckenhoupt weights $B_p$.

Characterization of the weak-type boundedness of the Hilbert transform on weighted Lorentz spaces

Abstract

We characterize the weak-type boundedness of the Hilbert transform on weighted Lorentz spaces , with , in terms of some geometric conditions on the weights and and the weak-type boundedness of the Hardy-Littlewood maximal operator on the same spaces. Our results recover simultaneously the theory of the boundedness of on weighted Lebesgue spaces and Muckenhoupt weights , and the theory on classical Lorentz spaces and Ariño Muckenhoupt weights .
Paper Structure (6 sections, 23 theorems, 112 equations)

This paper contains 6 sections, 23 theorems, 112 equations.

Table of Contents

  1. Introduction and motivation
  2. Several classes of weights
  3. The $B^*_\infty$ class
  4. The $B_{p, \infty}$ class
  5. $u\in A_\infty$ and $w\in B^*_\infty$
  6. Main Results

Key Result

Theorem 1.1

For every $0<p<\infty$, is bounded if and only if the following conditions hold: (i) $u\in A_\infty$. (ii) $w\in B^*_\infty$. (iii) $M: \Lambda^p_u(w) \to \Lambda^{p, \infty}_u(w)$ is bounded.

Theorems & Definitions (42)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Proposition 1.4
  • Definition 1.5
  • Proposition 1.6
  • Proposition 1.7
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • ...and 32 more