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Spectral multipliers for Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates. (English) Zbl 1221.42024
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 selfadjoint operator on $L^{2}(X)$ and $E(\lambda)$ the spectral resolution of $L$. For any bounded Borel function $m: [0,\infty)\to {\Bbb{C}}$, define $$m(L)=\int_0^{\infty}m(\lambda)\,dE(\lambda).$$ Assume that the semigroup $e^{-tL}$ generated by $L$ satisfies the Davies-Gaffney estimates, that is, there exist constants $C,c>0$, such that, for any open subset $U_1, U_2 \subset X$, $$|\langle e^{-tL}f_1,f_2\rangle|\le C\exp{\bigg(-\frac{\text{dist}(U_1,U_2)^2}{ct}\bigg)} \|f_1\|_{L^2(X)}\|f_2\|_{L^2(X)}, \quad \forall t>0,$$ for every $f_i\in L^2(X)$ with supp$f_i\subset U_i$, $i=1,2$. Denote by $H^p_L(X)$ the Hardy space associated with $L$. The authors establish a Hörmander-type spectral multiplier theorem for $L$ on $H^p_L(X)$ for $0 < p < \infty$. Precisely, let $\varphi$ be a non-negative $C_0^{\infty}$ function on ${\Bbb R}$ such that $$\operatorname {supp} \varphi\subseteq \bigg(\frac1{4},1\bigg) \quad \text{and} \quad \sum_{\ell\in {\Bbb{Z}}}\varphi(2^{-\ell}\lambda)=1 \quad \forall \lambda>0.$$ If $0<p\le 1$ and the bounded measurable function $m: [0,\infty)\to {\Bbb{C}}$ satisfies $$C_{\varphi,s}=\sup_{t>0}\|\varphi(\cdot)m(t\cdot)\|_{C^s} +|m(0)|<\infty$$ for some $s>n(1/p-1/2)$, then $m(L)$ is bounded from $H^p_L(X)$ to $H^p_L(X)$. If $C_{\varphi,s}<\infty$ for all $s>0$, then $m(L)$ is bounded on $H^p_L(X)$ for all $0 < p < \infty$. They also obtain a spectral multiplier theorem on $L^p$ spaces with appropriate weights in the reverse Hölder class.

##### MSC:
 42B20 Singular and oscillatory integrals, several variables 42B35 Function spaces arising in harmonic analysis 47B38 Operators on function spaces (general)
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##### References:
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