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Boundedness of Positive Integral Operators on Lorentz-Gamma Spaces

R. Kerman, S. Spektor

Abstract

We characterize the boundedness of a positive integral operator $T_K$, with kernel $K\in M_+(\R^{2n})$, between Lorentz-Gamma spaces $Γ_{p,φ_2}(\R^n)$ and $Γ_{q,φ_1}(\R^n)$, $1<p\le q<\infty$. The key step reduces the $n$-dimensional problem to a one-dimensional weighted norm inequality for the composed operator $T_LS$, where $L=(K^{*_2})^{*_1}$ is the iterated rearrangement of $K$ introduced by Blozinski~\cite{B} and $S$ is the Stieltjes transform. Explicit Muckenhoupt-type conditions are obtained for the case $L(t,s)=(t+s)^{-1}$, corresponding to the iterated Stieltjes operator $S^2$.

Boundedness of Positive Integral Operators on Lorentz-Gamma Spaces

Abstract

We characterize the boundedness of a positive integral operator , with kernel , between Lorentz-Gamma spaces and , . The key step reduces the -dimensional problem to a one-dimensional weighted norm inequality for the composed operator , where is the iterated rearrangement of introduced by Blozinski~\cite{B} and is the Stieltjes transform. Explicit Muckenhoupt-type conditions are obtained for the case , corresponding to the iterated Stieltjes operator .
Paper Structure (5 sections, 5 theorems, 29 equations)

This paper contains 5 sections, 5 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main Result
  4. Proof of Theorem \ref{['TH1N']}
  5. Application: the case $L(t,s)=(t+s)^{-1}$

Key Result

Theorem 3.1

Fix the indices $p$ and $q$, $1<p\leq q< \infty$. Let $\phi_1, \phi_2 \in M_+({\mathbb R}_+)$ be locally-integrable on ${\mathbb R}_+$. Set Suppose and Denote the dual weights of $\phi_1^{(q)}$ and $\phi_2$ by $\psi_1^{(q)}$ and $\psi_2$, respectively, so that, for example, Consider $K \in M_+({\mathbb R}^{2n})$ and define Then (1.1) holds whenever (1.2) does. Moreover, the latter is the case

Theorems & Definitions (8)

  • Theorem 3.1
  • Proposition 4.1
  • Remark 4.2
  • Proposition 4.3
  • Remark 5.1
  • Example 5.2
  • Theorem 5.3: BK
  • Corollary 5.4