Decay estimates for a class of semigroups related to self-adjoint operators on metric measure spaces

TL;DR

This paper establishes decay estimates for a broad class of dispersive semigroups related to self-adjoint operators on metric measure spaces, extending known results for Schrödinger operators to more general settings and applications.

Contribution

It introduces a method to derive decay estimates for general dispersive semigroups using subordination formulas, broadening the scope of analysis on metric measure spaces.

Findings

Established $L^1\to L^\infty$ decay estimates for $e^{it\phi(L)}$

Derived new Strichartz estimates for equations involving Hermite, twisted Laplacian, and Laguerre operators

Connected decay estimates with geometric properties of metric measure spaces

Abstract

Assume that $(X,d,\mu)$ is a metric space endowed with a non-negative Borel measure $\mu$ satisfying the doubling condition and the additional condition that $\mu(B(x,r))\gtrsim r^n$ for any $x\in X, \,r>0$ and some $n\geq1$. Let $L$ be a non-negative self-adjoint operator on $L^2(X,\mu)$. We assume that $e^{-tL}$ satisfies a Gaussian upper bound and the Schr\"odinger operator $e^{itL}$ satisfies an $L^1\to L^\infty$ decay estimate of the form \begin{equation*} \|e^{itL}\|_{L^1\to L^\infty} \lesssim |t|^{-\frac{n}{2}}. \end{equation*} Then for a general class of dispersive semigroup $e^{it\phi(L)}$, where $\phi: \mathbb{R}^+ \to \mathbb{R}$ is smooth, we establish a similar $L^1\to L^\infty$ decay estimate by a suitable subordination formula connecting it with the Schr\"odinger operator $e^{itL}$. As applications, we derive new Strichartz estimates for several dispersive equations related to Hermite operators, twisted Laplacians and Laguerre operators.