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Boundedness of spectral multipliers on locally compact groups and applications

Santiago Gómez Cobos, Joel E. Restrepo, Michael Ruzhansky

Abstract

We prove that the noncommutative Lorentz norm (associated to a semifinite von Neumann algebra) of a propagator of the form $\varphi(|\mathscr{L}|)$ can be estimated if the modulus of the Borel function $\varphi$ is bounded by a continuous positive monotonically decreasing function that vanishes at infinity $ψ$. As a consequence, we obtain the $L^p-L^q$ $(1<p\leqslant 2\leqslant q<+\infty)$ norm estimates for the solutions of heat, wave, and Schrödinger type equations (new in this setting) on a locally compact separable unimodular group $G$ by using a non-local integro-differential operator in time and any positive left invariant operator (maybe unbounded and with discrete or continuous spectrum) on $G$. We also provide asymptotic estimates (large-time behavior) for the solutions, which in some cases can be claimed to be sharp. Illustrative examples are given for several groups.

Boundedness of spectral multipliers on locally compact groups and applications

Abstract

We prove that the noncommutative Lorentz norm (associated to a semifinite von Neumann algebra) of a propagator of the form can be estimated if the modulus of the Borel function is bounded by a continuous positive monotonically decreasing function that vanishes at infinity . As a consequence, we obtain the norm estimates for the solutions of heat, wave, and Schrödinger type equations (new in this setting) on a locally compact separable unimodular group by using a non-local integro-differential operator in time and any positive left invariant operator (maybe unbounded and with discrete or continuous spectrum) on . We also provide asymptotic estimates (large-time behavior) for the solutions, which in some cases can be claimed to be sharp. Illustrative examples are given for several groups.
Paper Structure (12 sections, 6 theorems, 67 equations, 3 figures)

This paper contains 12 sections, 6 theorems, 67 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Abstract integro-differential equations
  4. Integro-differential operators
  5. Integro-differential equations
  6. Von Neumann algebras
  7. Main result and some applications
  8. L-heat type equation
  9. L-wave type equation
  10. L-Schrödinger type equation
  11. Examples
  12. Acknowledgements

Key Result

Theorem 1.1

Let $L$ be a closed (maybe unbounded) operator affiliatedSee the preliminaries for the definition. with a semifinite von Neuman algebra $M$. Let $\phi:[0,+\infty)\to\mathbb{C}$ be a Borel measurable function. Suppose also that $\psi$ is a monotonically decreasing continuous function on $[0,+\infty)$

Figures (3)

  • Figure 1: Wave propagators functions for $\alpha=1.95$ bounded uniformly.
  • Figure 2: Classical wave propagator function (green) is not uniformily bounded by a decreasing vanishing at infinity function.
  • Figure 3: Mittag-Leffler functions with one small parameter.

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 12 more