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April, 2013 $L^p$-bounds for Stein's square functions associated to operators and applications to spectral multipliers
Peng CHEN, Xuan Thinh DUONG, Lixin YAN
J. Math. Soc. Japan 65(2): 389-409 (April, 2013). DOI: 10.2969/jmsj/06520389


Let $(X, d, \mu)$ be a metric measure space endowed with a metric $d$ and a nonnegative Borel doubling measure $\mu$. Let $L$ be a non-negative self-adjoint operator of order $m$ on $X$. Assume that $L$ generates a holomorphic semigroup $e^{-tL}$ whose kernels $p_t(x,y)$ satisfy Gaussian upper bounds but without any assumptions on the regularity of space variables $x$ and $y$. Also assume that $L$ satisfies a Plancherel type estimate. Under these conditions, we show the $L^p$ bounds for Stein's square functions arising from Bochner-Riesz means associated to the operator $L$. We then use the $L^p$ estimates on Stein's square functions to obtain a Hörmander-type criterion for spectral multipliers of $L$. These results are applicable for large classes of operators including sub-Laplacians acting on Lie groups of polynomial growth and Schrödinger operators with rough potentials.


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Peng CHEN. Xuan Thinh DUONG. Lixin YAN. "$L^p$-bounds for Stein's square functions associated to operators and applications to spectral multipliers." J. Math. Soc. Japan 65 (2) 389 - 409, April, 2013.


Published: April, 2013
First available in Project Euclid: 25 April 2013

zbMATH: 1277.42011
MathSciNet: MR3055591
Digital Object Identifier: 10.2969/jmsj/06520389

Primary: 42B15
Secondary: 42B25 , 47F05

Keywords: Bochner-Riesz means , heat semigroup , non-negative self-adjoint operator , space of homogeneous type , Stein's square function

Rights: Copyright © 2013 Mathematical Society of Japan


Vol.65 • No. 2 • April, 2013
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