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Generalized Carleson Embeddings of M{ü}ntz Spaces

Mickaël Latocca, Vincent Munnier

Abstract

This paper establishes Carleson embeddings of M{ü}ntz spaces $M^q_Λ$ into weighted Lebesgue spaces $L^p(\mathrm{d}μ)$, where $μ$ is a Borel regular measure on $[0,1]$ satisfying $μ([1-\varepsilon])\lesssim \varepsilon^β$. In the case $β\geqslant 1$ we show that such measures are exactly the ones for which Carleson embeddings $L^{\frac{p}β} \hookrightarrow L^p(\mathrm{d}μ)$ hold. The case $β\in (0,1)$ is more intricate but we characterize such measures $μ$ in terms of a summability condition on their moments. Our proof relies on a generalization of $L^p$ estimates {à} la Gurariy-Macaev in the weighted $L^p$ spaces setting, which we think can be of interest in other contexts.

Generalized Carleson Embeddings of M{ü}ntz Spaces

Abstract

This paper establishes Carleson embeddings of M{ü}ntz spaces into weighted Lebesgue spaces , where is a Borel regular measure on satisfying . In the case we show that such measures are exactly the ones for which Carleson embeddings hold. The case is more intricate but we characterize such measures in terms of a summability condition on their moments. Our proof relies on a generalization of estimates {à} la Gurariy-Macaev in the weighted spaces setting, which we think can be of interest in other contexts.
Paper Structure (16 sections, 19 theorems, 131 equations)

This paper contains 16 sections, 19 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Carleson Embeddings
  3. Main results
  4. Plan of the paper
  5. Preliminaries
  6. Notation
  7. Useful properties of Müntz polynomials
  8. Bernstein inequalities
  9. The non-singular case: proof of Theorem \ref{['thm:non-sing']}
  10. The singular case: proof of Theorem \ref{['thm:sing']}
  11. The decoupling estimates: proof of Theorem \ref{['thm:Lp-frames']}
  12. The non-singular case
  13. The singular case
  14. The multilinear estimate: proof of Theorem \ref{['thm:multilinear']}
  15. The bilinear case
  16. ...and 1 more sections

Key Result

Theorem A

Let $\mu$ be a positive Borel measure on $[0,1]$ and let $\beta\geqslant 1$. Let also $\Lambda$ be a quasi-lacunary and subgeometric sequence with parameters $q, N$. Then the following assertions are equivalent:

Theorems & Definitions (41)

  • Theorem A: Non-singular case
  • Theorem B: Singular Case
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem C: $L^p$ decoupling estimates
  • Remark 1.5
  • Theorem D: Multilinear estimate
  • Remark 1.6
  • ...and 31 more