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arXiv · 1211.1295

Sharp spectral multipliers for operators satisfying generalized Gaussian estimates

Adam Sikora·Lixin Yan·Xiaohua Yao
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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.

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Adam Sikora, Lixin Yan, Xiaohua Yao. 2012-11-06. Sharp spectral multipliers for operators satisfying generalized Gaussian estimates. https://arxiv.org/abs/1211.1295

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