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$H^1$ to $L^q$-boundedness of fractional integral operators having a flag kernel

Jiashu Zhang, Zipeng Wang

TL;DR

This work characterizes the $H^1(\mathbb{R}^{n+m}) \to L^q(\mathbb{R}^{n+m})$ boundedness of a family of flag-kernel fractional integral operators $I_{αβ}^{ρ}$ with kernels $Ω^{αβ}_{ρ}$ under non-isotropic dilations. It identifies necessary dilation-based conditions for general $L^p \to L^q$ boundedness, and shows a sharp endpoint result: $I_{αβ}^{ρ}$ maps ${H^1}$ to ${L^q}$ when $α/n > β/m$ and $(α+ρβ)/(n+ρ m) = 1 - 1/q$, while a counterexample demonstrates failure at the borderline $α/n = β/m$. The proofs combine kernel-dominance arguments and Hardy-Littlewood-Sobolev theory for the $L^p \to L^q$ case with an atomic-decomposition/dyadic analysis for the $H^1$ endpoint, exploiting the flag structure of the kernel. Overall, the results extend classical and product Hardy space theory to non-isotropic, flag-type fractional integrals and precisely delineate end-point behavior.

Abstract

We study a family of fractional integral operators whose kernels satisfying an non-isotropic dilation have singularity on a coordinate subspace. A characterization is given for these operators bounded from the classical, atom decomposable $H^1$-Hardy space to $L^q$-spaces.

$H^1$ to $L^q$-boundedness of fractional integral operators having a flag kernel

TL;DR

This work characterizes the boundedness of a family of flag-kernel fractional integral operators with kernels under non-isotropic dilations. It identifies necessary dilation-based conditions for general boundedness, and shows a sharp endpoint result: maps to when and , while a counterexample demonstrates failure at the borderline . The proofs combine kernel-dominance arguments and Hardy-Littlewood-Sobolev theory for the case with an atomic-decomposition/dyadic analysis for the endpoint, exploiting the flag structure of the kernel. Overall, the results extend classical and product Hardy space theory to non-isotropic, flag-type fractional integrals and precisely delineate end-point behavior.

Abstract

We study a family of fractional integral operators whose kernels satisfying an non-isotropic dilation have singularity on a coordinate subspace. A characterization is given for these operators bounded from the classical, atom decomposable -Hardy space to -spaces.
Paper Structure (9 sections, 78 equations, 1 figure)

This paper contains 9 sections, 78 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof of necessary conditions
  3. A counter example for ${\bf H}^1\longrightarrow{\bf L}^q$-estimate
  4. Proof of Theorem One
  5. Proof of Theorem Two
  6. Case 1: $k=\ell=0$
  7. Case 2: $k>0$, $\ell>0$
  8. Case 3: $k>0$, $\ell=0$ or $k=0$, $\ell>0$
  9. Case 4: $k=0$ and $\ell>0$

Figures (1)

  • Figure :

Theorems & Definitions (2)

  • Remark 1.1
  • Remark 1.2