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Spectral multipliers III: Endpoint bounds, intertwining operators, and twisted Hardy spaces

Marius Beceanu, Michael Goldberg

Abstract

We extend several fundamental estimates regarding spectral multipliers for the free Laplacian on $\mathbb R^3$ to the case of perturbed Hamiltonians of the form $H=-Δ+V$, where $V$ is a scalar real-valued potential. Results include sharp bounds for Mihlin multipliers, partial confirmation for a conjecture made in [BeGo3] about intertwining operators, a characterization of the twisted Hardy spaces that correspond to these perturbed Hamiltonians, Strichartz estimates, and maximum principles.

Spectral multipliers III: Endpoint bounds, intertwining operators, and twisted Hardy spaces

Abstract

We extend several fundamental estimates regarding spectral multipliers for the free Laplacian on to the case of perturbed Hamiltonians of the form , where is a scalar real-valued potential. Results include sharp bounds for Mihlin multipliers, partial confirmation for a conjecture made in [BeGo3] about intertwining operators, a characterization of the twisted Hardy spaces that correspond to these perturbed Hamiltonians, Strichartz estimates, and maximum principles.
Paper Structure (12 sections, 30 theorems, 195 equations)

This paper contains 12 sections, 30 theorems, 195 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. The twisted Hardy space
  4. Paley--Wiener decompositions and square functions
  5. Strichartz estimates for the wave equation
  6. Notations
  7. Preliminaries
  8. Maximum principles
  9. Mihlin multipliers
  10. The twisted Hardy space
  11. Intertwining operators
  12. The wave equation

Key Result

Theorem 1.1

Assume $V \in \mathcal{K}_0$, and $H = -\Delta + V$ has no eigenvalue or resonance at zero energy, and no positive eigenvalues. Let $\phi$ be a standard cutoff function such that $\phi \in C^\infty$, $\mathop{\mathrm{supp}}\nolimits \phi \subset [\frac{1}{2}, 4]$, and $\phi(x)=1$ for $x \in [1, 2]$. or equivalently $\sup_{\alpha>0} \|\phi(\lambda)m(\alpha^{-1} \lambda)\|_{H^s} < \infty$. Then $m(\

Theorems & Definitions (58)

  • Theorem 1.1: Main Theorem
  • Remark 1.2
  • Definition 1.1
  • Proposition 1.3
  • Theorem 1.4
  • Lemma 1.5
  • Theorem 1.6
  • Lemma 1.7: hong
  • Definition 1.2
  • Definition 1.3
  • ...and 48 more