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Quantitative Weighted Estimates for Schrödinger Pseudo-Multipliers and its Commutators

Sayan Bagchi, Riju Basak, Joydwip Singh, Manasa N. Vempati

Abstract

In this article, we investigate the unweighted and weighted $L^p$-boundedness of pseudo-multipliers associated with a class of Schrödinger operators. The weight classes we consider are tailored to this framework and strictly contain the classical Muckenhoupt $A_p$-classes. To establish the weighted boundedness, we prove a quantitative version of reverse Hölder's inequality and quantitative weighted estimates for general sparse operators, which are of independent interest. We also study commutators of Schrödinger pseudo-multipliers, establishing their boundedness and compactness results on these weighted $L^p$-spaces.

Quantitative Weighted Estimates for Schrödinger Pseudo-Multipliers and its Commutators

Abstract

In this article, we investigate the unweighted and weighted -boundedness of pseudo-multipliers associated with a class of Schrödinger operators. The weight classes we consider are tailored to this framework and strictly contain the classical Muckenhoupt -classes. To establish the weighted boundedness, we prove a quantitative version of reverse Hölder's inequality and quantitative weighted estimates for general sparse operators, which are of independent interest. We also study commutators of Schrödinger pseudo-multipliers, establishing their boundedness and compactness results on these weighted -spaces.
Paper Structure (16 sections, 26 theorems, 209 equations)

This paper contains 16 sections, 26 theorems, 209 equations.

Table of Contents

  1. Introduction and Main results
  2. Schrödinger pseudo-multiplier
  3. Examples:
  4. Preliminaries
  5. Sparse operators and reverse Hölder's inequality
  6. Kernel estimates
  7. boundedness of Schrödinger pseudo-multipliers
  8. Quantitative weighted -estimates for the Schrödinger pseudo-multipliers
  9. Estimate of
  10. Estimate of
  11. Quantitative weighted -estimates for commutators of the Schrödinger pseudo-multipliers
  12. Estimate of
  13. Estimate of
  14. Estimate of
  15. Estimate of
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\sigma \in S^0_{1,0}(\mathbb{R}^n)$. Then for $1\leq r <p<\infty$ and $\omega \in A_{p/r}$, there exists a constant $C>0$ independent of $\omega$ such that

Theorems & Definitions (43)

  • Theorem 1.1
  • Definition 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Lemma 2.1
  • Remark 2.1
  • Proposition 2.1
  • ...and 33 more