Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Boundedness of the Hilbert transform on weighted Lorentz spaces

Elona Agora, María J. Carro, Javier Soria

Abstract

We study the boundedness of the Hilbert transform $H$ and the Hilbert maximal operator $H^*$ on weighted Lorentz spaces $Λ^p_u(w)$. We start by giving several necessary conditions that, in particular, lead us to the complete characterization of the weak-type boundedness of both $H$ and $H^*$, whenever $u\in A_1$. For the strong-type case, we also get the characterization of both operators when $p>1$. Applications to the case of Lorentz spaces $L^{p,q}(u)$ are presented.

Boundedness of the Hilbert transform on weighted Lorentz spaces

Abstract

We study the boundedness of the Hilbert transform and the Hilbert maximal operator on weighted Lorentz spaces . We start by giving several necessary conditions that, in particular, lead us to the complete characterization of the weak-type boundedness of both and , whenever . For the strong-type case, we also get the characterization of both operators when . Applications to the case of Lorentz spaces are presented.
Paper Structure (3 sections, 16 theorems, 107 equations)

This paper contains 3 sections, 16 theorems, 107 equations.

Table of Contents

  1. Introduction and motivation
  2. Necessary conditions
  3. Consequences and Applications

Key Result

Proposition 1.1

(a) $\Lambda^{p}_{u}(w)$ and $\Lambda^{p,\infty}_{u}(w)$ are quasi-normed spaces if and only if $w$ satisfies the $\Delta_2$ condition ($w\in \Delta_2$); that is, $W(2r)\lesssim W(r)$, $r>0$. (b) Assume that $u\notin L^1(\mathbb R)$, $w\notin L^1(\mathbb R^+)$ and $w\in \Delta_2$. If $|g_n|\le |f|$, is dense in $\Lambda^p_u(w)$.

Theorems & Definitions (35)

  • Proposition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 25 more