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Boundedness and norm of certain p-adic Hardy-Littlewood-Pólya-type operators

Jianjun Jin, Huabing Li

Abstract

In this paper, by introducing some parameters, we define and study certain $p$-adic Hardy-Littlewood-Pólya-type integral operators acting on $p$-adic weighted Lebesgue spaces. We completely characterize $L^{q}-L^{r}$ boundedness of these operators for all $(q, r)\in [1, \infty]\times[1, \infty]$. For some special cases, we obtain sharp norm estimates for the operators. These results are not only a complement to some previous results but also an extension of existing ones in the literature.

Boundedness and norm of certain p-adic Hardy-Littlewood-Pólya-type operators

Abstract

In this paper, by introducing some parameters, we define and study certain -adic Hardy-Littlewood-Pólya-type integral operators acting on -adic weighted Lebesgue spaces. We completely characterize boundedness of these operators for all . For some special cases, we obtain sharp norm estimates for the operators. These results are not only a complement to some previous results but also an extension of existing ones in the literature.
Paper Structure (11 sections, 13 theorems, 172 equations)

This paper contains 11 sections, 13 theorems, 172 equations.

Table of Contents

  1. Introduction and main results
  2. Some lemmas
  3. Proof of Theorem \ref{['th-1']}
  4. Case I. $1<q\leq r<\infty$
  5. Case II. $1=q\leq r<\infty$
  6. Proof of Theorem \ref{['th-2']} and \ref{['th-3']}
  7. Proof of Theorem \ref{['th-2']}
  8. Proof of Theorem \ref{['th-3']}
  9. Proof of Theorem \ref{['th-4']}
  10. Proof of Theorem \ref{['th-5']}
  11. Final remarks

Key Result

Theorem 1.1

Let $q>1$. Let $H$ be as above. Then $H$ is bounded on $L^q(\mathbb{R}_+)$ and the norm $\|H\|_{L^q\rightarrow L^q}$ of $H$ is $q+q'$. Here

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Lemma 2.1
  • Remark 2.2
  • ...and 14 more