Lixin Yan
Restriction estimates, sharp spectral multipliers and endpoint estimates for Bochner-Riesz means.
Tue, 02/21/2012 - 15:01 — SomNambulAuthors: Adam Sikora, Lixin Yan, Peng Chen, El Maati Ouhabaz
Subjects: Analysis of PDEs
AbstractWe consider abstract non-negative self-adjoint operators on $L^2(X)$ which
satisfy the finite speed propagation property for the corresponding wave
equation. For such operators we introduce a restriction type condition which in
the case of the standard Laplace operator is equivalent to $(p,2)$ restriction
estimate of Stein and Tomas. Next we show that in the considered abstract
setting our restriction type condition implies sharp spectral multipliers and
endpoint estimates for the Bochner-Riesz summability.Weighted norm inequalities, Gaussian bounds and sharp spectral multipliers.
Wed, 03/10/2010 - 16:01 — SomNambulAuthors: Adam Sikora, Xuan Thinh Duong, Lixin Yan
Subjects: Functional Analysis
AbstractLet $L$ 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 $p_t(x,y)$ have Gaussian upper bounds but
possess no regularity in variables $x$ and $y$. In this article, we study
weighted $L^p$-norm inequalities for spectral multipliers of $L$. We show sharp
weighted H\"ormander-type spectral multiplier theorems follow from Gaussian
heat kernel bounds and appropriate $L^2$ estimates of the kernels of the
spectral multipliers.
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