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Two-weight dyadic Hardy's inequalities

Nicola Arcozzi, Nikolaos Chalmoukis, Matteo Levi, Pavel Mozolyako

Abstract

We present various results concerning the two-weight Hardy's inequality on infinite trees. Our main scope is to survey known characterizations (and proofs) for trace measures, as well as to provide some new ones. Also for some of the known characterizations we provide here new proofs. In particular, we obtain a new characterization based on a new reverse Hölder inequality for trace measures, and one based on the well known Muckenhoupt-Wheeden-Wolff inequality, of which we here give a new probabilistic proof. We provide a new direct proof for the so called isocapacitary characterization and a new simple proof, based on a monotonicity argument, for the so called mass-energy characterization. Furthermore, we introduce a conformally invariant version of the two-weight Hardy's inequality, we characterize the compactness of the Hardy operator, we provide a list of open problems and suggest some possible lines of future research.

Two-weight dyadic Hardy's inequalities

Abstract

We present various results concerning the two-weight Hardy's inequality on infinite trees. Our main scope is to survey known characterizations (and proofs) for trace measures, as well as to provide some new ones. Also for some of the known characterizations we provide here new proofs. In particular, we obtain a new characterization based on a new reverse Hölder inequality for trace measures, and one based on the well known Muckenhoupt-Wheeden-Wolff inequality, of which we here give a new probabilistic proof. We provide a new direct proof for the so called isocapacitary characterization and a new simple proof, based on a monotonicity argument, for the so called mass-energy characterization. Furthermore, we introduce a conformally invariant version of the two-weight Hardy's inequality, we characterize the compactness of the Hardy operator, we provide a list of open problems and suggest some possible lines of future research.

Paper Structure

This paper contains 25 sections, 29 theorems, 189 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A (very) brief history of Hardy's inequality
  3. The two-weight Hardy's inequality on trees
  4. Description of contents
  5. Potential theory on trees and the isocapacitary characterization
  6. Potential theory
  7. Strong Capacitary Inequality and the isocapacitary characterization
  8. Mass--energy characterization: three different proofs
  9. Maximal function
  10. Monotone proof for p=2
  11. Bellman function
  12. A reverse Hölder inequality
  13. The inequality of Muckenhoupt and Wheeden, and Wolff
  14. Conformally invariant Hardy's inequality
  15. Miscellaneous results
  16. ...and 10 more sections

Key Result

Theorem 1

Let $1\leq p \leq \infty$. There exists a constant $C$ such that Continuous Hardy is true if and only if Furthermore, if $C$ is the smallest constant such that the inequality holds, then $B\leq C\leq p (p^*)^{p-1} B.$

Figures (2)

  • Figure 1: The infinite rooted tree.
  • Figure 2: A snapshot from the defined family of points and their relations: continuous lines represent edges and dashed lines paths of (many) edges. Next to some vertices is specified, in parenthesis, their distance from the origin.

Theorems & Definitions (32)

  • Theorem 1: Muckenhoupt
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5: Capacitary Strong Inequality
  • Theorem 6
  • Lemma 7
  • Lemma 8
  • Theorem 9: Carleson imbedding theorem for trees
  • Theorem 10
  • ...and 22 more