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Vector-Valued Singular Integrals on Locally Doubling Spaces

Mattia Calzi, Elena Rizzo

TL;DR

The paper develops a local Calderón–Zygmund theory for vector-valued operators on connected locally doubling spaces. It shows that under radius-controlled doubling and kernel smoothness, localized operators satisfy $L^p$ bounds for $p\in(1,r]$ with explicit dependence on a Marcinkiewicz-type interpolation factor $\varphi(r,p)$ and local constants, e.g., $ rm{T f}_{L^p(\mu;B_2)} \le 16\,\varphi(r,p) (D_{3\kappa R}^9 A_R + C_R) \nrm{f}_{L^p(\mu;B_1)}$. A local-to-global patching argument then yields global-type estimates, and the vector-valued maximal operator on mixed-norm spaces is likewise bounded with constants depending on $D_R$ and $\varphi$. These results extend classical doubling-space theory to locally doubling spaces, with applications to BesovTL and related analyses.

Abstract

We prove vector-valued boundedness of (suitable) Calderon-Zygmund operators and of the (truncated) Hardy-Littlewood maximal function on a connected locally doubling metric measure space.

Vector-Valued Singular Integrals on Locally Doubling Spaces

TL;DR

The paper develops a local Calderón–Zygmund theory for vector-valued operators on connected locally doubling spaces. It shows that under radius-controlled doubling and kernel smoothness, localized operators satisfy bounds for with explicit dependence on a Marcinkiewicz-type interpolation factor and local constants, e.g., . A local-to-global patching argument then yields global-type estimates, and the vector-valued maximal operator on mixed-norm spaces is likewise bounded with constants depending on and . These results extend classical doubling-space theory to locally doubling spaces, with applications to BesovTL and related analyses.

Abstract

We prove vector-valued boundedness of (suitable) Calderon-Zygmund operators and of the (truncated) Hardy-Littlewood maximal function on a connected locally doubling metric measure space.
Paper Structure (2 sections, 12 theorems, 102 equations)

This paper contains 2 sections, 12 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Statement and Proof of the Main Results

Key Result

Lemma 2

Take $R>0$ so that $D_R<\infty$. Then $\mu(U), \mu(B(x,r))\in (0,+\infty)$ for every non-empty open subset $U$ of $X$ with $\mathop{\mathrm{diam}}\nolimits(U)<2R$, for every $x\in X$, and for every $r\in (0,2R]$.

Theorems & Definitions (34)

  • Remark 1
  • Lemma 2
  • proof
  • Remark 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Definition 6
  • Remark 7
  • ...and 24 more