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Weak and strong type $A_1$-$A_\infty$ estimates for sparsely dominated operators

Dorothee Frey, Zoe Nieraeth

Abstract

We consider operators $T$ satisfying a sparse domination property \[ |\langle Tf,g\rangle|\leq c\sum_{Q\in\mathscr{S}}\langle f\rangle_{p_0,Q}\langle g\rangle_{q_0',Q}|Q| \] with averaging exponents $1\leq p_0<q_0\leq\infty$. We prove weighted strong type boundedness for $p_0<p<q_0$ and use new techniques to prove weighted weak type $(p_0,p_0)$ boundedness with quantitative mixed $A_1$-$A_\infty$ estimates, generalizing results of Lerner, Ombrosi, and Pérez and Hytönen and Pérez. Even in the case $p_0=1$ we improve upon their results as we do not make use of a Hörmander condition of the operator $T$. Moreover, we also establish a dual weak type $(q_0',q_0')$ estimate. In a last part, we give a result on the optimality of the weighted strong type bounds including those previously obtained by Bernicot, Frey, and Petermichl.

Weak and strong type $A_1$-$A_\infty$ estimates for sparsely dominated operators

Abstract

We consider operators satisfying a sparse domination property with averaging exponents . We prove weighted strong type boundedness for and use new techniques to prove weighted weak type boundedness with quantitative mixed - estimates, generalizing results of Lerner, Ombrosi, and Pérez and Hytönen and Pérez. Even in the case we improve upon their results as we do not make use of a Hörmander condition of the operator . Moreover, we also establish a dual weak type estimate. In a last part, we give a result on the optimality of the weighted strong type bounds including those previously obtained by Bernicot, Frey, and Petermichl.

Paper Structure

This paper contains 10 sections, 18 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. The setting
  3. Main results
  4. Examples
  5. Preliminaries
  6. Notation
  7. Weighted boundedness of sparsely dominated operators
  8. Proofs of the main results
  9. Extensions of the results to spaces of homogeneous type
  10. Optimality of weighted strong type estimates

Key Result

Theorem 1.3

Let $1\leq p_0<p<q_0\leq\infty$, $T\in S(p_0,q_0)$, and $w\in A_1\cap\mathop{\mathrm{RH}}\nolimits_{(q_0/p)'}$. Then there is a constant $c=c(T,\nu,n)>0$ so that with In particular, we have

Theorems & Definitions (44)

  • Remark 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Example 1.7: Riesz transform associated with elliptic second order divergence form operators
  • Example 1.8: Riesz transform associated to Neumann Laplacian
  • Example 1.9: Fourier multipliers
  • Proposition 2.1
  • ...and 34 more