Journal of the Mathematical Society of Japan

$L^p$-bounds for Stein's square functions associated to operators and applications to spectral multipliers

Peng CHEN, Xuan Thinh DUONG, and Lixin YAN

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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.

Article information

J. Math. Soc. Japan, Volume 65, Number 2 (2013), 389-409.

First available in Project Euclid: 25 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B15: Multipliers
Secondary: 42B25: Maximal functions, Littlewood-Paley theory 47F05: Partial differential operators [See also 35Pxx, 58Jxx] (should also be assigned at least one other classification number in section 47)

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


CHEN, Peng; DUONG, Xuan Thinh; YAN, Lixin. $L^p$-bounds for Stein's square functions associated to operators and applications to spectral multipliers. J. Math. Soc. Japan 65 (2013), no. 2, 389--409. doi:10.2969/jmsj/06520389.

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