Journal of the Mathematical Society of Japan

Spectral multipliers for Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates

Xuan Thinh DUONG and Lixin YAN

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Let $(X,d,\mu)$ be a metric measure space endowed with a distance $d$ and a nonnegative Borel doubling measure $\mu$. Let $L$ be a non-negative self-adjoint operator on $L^2(X)$. Assume that the semigroup $e^{-tL}$ generated by $L$ satisfies the Davies-Gaffney estimates. Let $H_L^p(X)$ be the Hardy space associated with $L$. We prove a Hörmander-type spectral multiplier theorem for $L$ on $H_L^p(X)$ for $0 < p <\infty:$ the operator $m(L)$ is bounded from $H_L^p(X)$ to $H_L^p(X)$ if the function $m$ possesses $s$ derivatives with suitable bounds and $s > n(1/p - 1/2)$ where $n$ is the "dimension" of $X$. By interpolation, $m(L)$ is bounded on $H_L^p(X)$ for all $0 < p < \infty$ if $m$ is infinitely differentiable with suitable bounds on its derivatives. We also obtain a spectral multiplier theorem on $L^p$ spaces with appropriate weights in the reverse Hölder class.

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J. Math. Soc. Japan Volume 63, Number 1 (2011), 295-319.

First available in Project Euclid: 27 January 2011

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Mathematical Reviews number (MathSciNet)

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B35: Function spaces arising in harmonic analysis
Secondary: 47B38: Operators on function spaces (general)

spectral multipliers Hardy space non-negative self-adjoint operators Davies-Gaffney estimate atom molecule space of homogeneous type


DUONG, Xuan Thinh; YAN, Lixin. Spectral multipliers for Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates. J. Math. Soc. Japan 63 (2011), no. 1, 295--319. doi:10.2969/jmsj/06310295.

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