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Endpoint estimates and sparse domination in nonhomogeneous trees

José M. Conde-Alonso, Filippo De Mari, Matteo Monti, Elena Rizzo, Maria Vallarino

Abstract

We prove endpoint and sparse-like bounds for Bergman projectors on nonhomogeneous, radial trees $X$ that model manifolds with possibly unbounded geometry. The natural Bergman measures on $X$ may fail to be doubling, and even locally doubling, with respect to the right metric in our setting. Weighted consequences of our sparse domination results are also considered, and are in line with the known results in the disk. Our endpoint results are partly a consequence of a new Calderón-Zygmund theory for discrete, non-locally doubling metric spaces.

Endpoint estimates and sparse domination in nonhomogeneous trees

Abstract

We prove endpoint and sparse-like bounds for Bergman projectors on nonhomogeneous, radial trees that model manifolds with possibly unbounded geometry. The natural Bergman measures on may fail to be doubling, and even locally doubling, with respect to the right metric in our setting. Weighted consequences of our sparse domination results are also considered, and are in line with the known results in the disk. Our endpoint results are partly a consequence of a new Calderón-Zygmund theory for discrete, non-locally doubling metric spaces.

Paper Structure

This paper contains 11 sections, 11 theorems, 149 equations.

Table of Contents

  1. Trees with unbounded geometry
  2. Radial trees of (possibly) unbounded degree
  3. Martingale structure. $\mathrm{H}^1$ and $\mathrm{BMO}$
  4. Operators acting on $\mathcal{X}$: Bergman spaces and projection-like operators
  5. Kernel estimates for $\mathcal{P}$
  6. Sparse forms associated with $\mathcal{P}$
  7. Sparse domination
  8. Discrete Bekollé-Bonami weights
  9. Endpoint estimates
  10. General Calderón-Zygmund theory in $\mathcal{X}$
  11. Endpoint results for $\mathcal{P}$

Key Result

Theorem 1

Let $f_1,f_2 \in L^1(\mathcal{X})$. There exists a sparse collection $\mathcal{S}$ such that

Theorems & Definitions (32)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Remark 1.1
  • Lemma 1.2
  • proof
  • Remark 1.3
  • Remark 1.4
  • Lemma 1.5
  • ...and 22 more