Sharp spectral multipliers for operators satisfying generalized Gaussian estimates

TL;DR

This paper establishes sharp spectral multiplier theorems for self-adjoint operators with kernels satisfying generalized Gaussian estimates, extending results to various classes of differential and fractal operators.

Contribution

It demonstrates that sharp spectral multiplier results can be derived from Plancherel or Stein-Tomas estimates in this generalized setting.

Findings

Spectral multipliers follow from Plancherel or Stein-Tomas estimates.

Results apply to elliptic, biharmonic, and fractal operators.

Provides a unified approach for operators with Gaussian kernel bounds.

Abstract

Let $L$ be a non-negative self adjoint operator acting on $L^2(X)$ where $X$ is a space of homogeneous type. Assume that $L$ generates a holomorphic semigroup $e^{-tL}$ whose kernels $p_t(x,y)$ satisfy generalized $m$-th order Gaussian estimates. In this article, we study singular and dyadically supported spectral multipliers for abstract self-adjoint operators. We show that in this setting sharp spectral multiplier results follow from Plancherel or Stein-Tomas type estimates. These results are applicable to spectral multipliers for large classes of operators including $m$-th order elliptic differential operators with constant coefficients, biharmonic operators with rough potentials and Laplace type operators acting on fractals.