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A weak to strong type T1 theorem for general smooth Calderón-Zygmund operators with doubling weights, II

Michel Alexis, Eric T. Sawyer, Ignacio Uriarte-Tuero

Abstract

We consider the weak to strong type problem for two weight norm inequalities for Calderón-Zygmund operators with doubling weights. We show that if a Calderón-Zygmund operator T is weak type (2,2) with doubling weights, then it is strong type (2,2) if and only if the dual cube testing condition for T^{*} holds, alternatively if and only if the dual cancellation condition of Stein holds. The testing condition can be taken with respect to either cubes or balls, and more generally, this is extended to a weak form of Tb theorem. Finally, we show that for all pairs of locally finite positive Borel measures, and all Stein elliptic Calderón-Zygmund operators T, the weak type (2,2) inequalities for T and and its associated maximal truncations operator T_{*} are equivalent. Thus the characterization of weak type for T_{*} in [LaSaUr1] applies to T as well.

A weak to strong type T1 theorem for general smooth Calderón-Zygmund operators with doubling weights, II

Abstract

We consider the weak to strong type problem for two weight norm inequalities for Calderón-Zygmund operators with doubling weights. We show that if a Calderón-Zygmund operator T is weak type (2,2) with doubling weights, then it is strong type (2,2) if and only if the dual cube testing condition for T^{*} holds, alternatively if and only if the dual cancellation condition of Stein holds. The testing condition can be taken with respect to either cubes or balls, and more generally, this is extended to a weak form of Tb theorem. Finally, we show that for all pairs of locally finite positive Borel measures, and all Stein elliptic Calderón-Zygmund operators T, the weak type (2,2) inequalities for T and and its associated maximal truncations operator T_{*} are equivalent. Thus the characterization of weak type for T_{*} in [LaSaUr1] applies to T as well.

Paper Structure

This paper contains 29 sections, 13 theorems, 232 equations.

Table of Contents

  1. Introduction
  2. Weak type inequalities
  3. Preliminaries
  4. Standard fractional singular integrals and the norm inequality
  5. Ellipticity of kernels
  6. Defining the norm inequality
  7. $\kappa$-cube testing conditions
  8. Weighted Alpert bases for $L^{2}\left( \mu\right)$ and $L^{\infty}$ control of projections
  9. The Pivotal Lemma
  10. The Calderón-Zygmund corona decomposition
  11. Parallel corona decompositions
  12. Reduction of the proof to local forms
  13. Cube Size Splitting
  14. Shifted Corona Decomposition
  15. The initial reduction using testing
  16. ...and 14 more sections

Key Result

Theorem 2

Suppose $\sigma$ and $\omega$ are doubling measures on $\mathbb{R}^{n}$, and that $T^{\alpha}$ is a smooth Stein elliptic Calderón-Zygmund operator on $\mathbb{R}^{n}$. Then (strong) holds if and only if the testing constants in (Muck and test) and (ind) are finite, and furthermore, we can replace ( and the corresponding equivalences with $T^{\alpha}$ and $T^{\alpha,\ast}$ and their constants inte

Theorems & Definitions (30)

  • Theorem 2
  • Remark 3
  • Corollary 4
  • Theorem 5
  • Theorem 6
  • Remark 7
  • Remark 8
  • Remark 9
  • Definition 10
  • Lemma 11: Saw6
  • ...and 20 more