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Sparse Domination of Singular Bilinear Forms on Non-Homogeneous spaces

Paco Villarroya

Abstract

We introduce a new sparse $T1$ theorem that estimates the dual pair associated with a Calderon-Zygmund operator by a sub-bilinear form supported on a sparse family of cubes. The main result in the paper improves previous sparse $T1$ theorems in several ways: it applies to non-homogeneous measures of power growth, it only requires a numerable family of testing conditions, and it can be used to prove boundedness of Calderon-Zygmund operators on weighted spaces for a class of weights larger than the Muckenhoupt $A_p$ weights.

Sparse Domination of Singular Bilinear Forms on Non-Homogeneous spaces

Abstract

We introduce a new sparse theorem that estimates the dual pair associated with a Calderon-Zygmund operator by a sub-bilinear form supported on a sparse family of cubes. The main result in the paper improves previous sparse theorems in several ways: it applies to non-homogeneous measures of power growth, it only requires a numerable family of testing conditions, and it can be used to prove boundedness of Calderon-Zygmund operators on weighted spaces for a class of weights larger than the Muckenhoupt weights.
Paper Structure (13 sections, 20 theorems, 214 equations)

This paper contains 13 sections, 20 theorems, 214 equations.

Table of Contents

  1. Introduction
  2. Notation, definitions, and statement of main result
  3. Measures of power growth
  4. Two Haar wavelet systems
  5. Construction of a sparse collection of stopping cubes
  6. Bump estimates, Paraproducts, and Square functions
  7. Estimates of the dual form by bilinear forms acting on square functions
  8. Paraproducts and essentially attached cubes.
  9. Paraproducts
  10. Essentially attached cubes and fragments of the two paraproducts
  11. An iteration process to end the proof of Proposition \ref{['dualbysquare']}
  12. Boundedness of the square functions on $L^2$ and weak $L^1$
  13. A recursion process to dominate the square forms by a sparse form

Key Result

Theorem 2.7

Let $\mu$ be a positive Radon measure on $\mathbb R^n$ with power growth $0<\alpha \leq n$. Let $k=n-\lfloor\alpha \rfloor+\delta(\alpha -\lfloor\alpha\rfloor)$. Let $T$ be a linear operator with a Calderón-Zygmund kernel as in smoothcompactCZ and kernelrep with parameter $0<\delta \leq 1$, and boun holds for all $I\in \mathcal{D}_i$ (only on a sparse family of cubes.) Then, for $f, g$ bounded wit

Theorems & Definitions (48)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 38 more