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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 $\left(X,d,\mu \right)$ 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 ${L}^{2}\left(X\right)$ and $E\left(\lambda \right)$ the spectral resolution of $L$. For any bounded Borel function $m:\left[0,\infty \right)\to ℂ$, define

$m\left(L\right)={\int }_{0}^{\infty }m\left(\lambda \right)\phantom{\rule{0.166667em}{0ex}}dE\left(\lambda \right)·$

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 ${U}_{1},{U}_{2}\subset X$,

$|〈{e}^{-tL}{f}_{1},{f}_{2}〉|\le Cexp\left(-\frac{\text{dist}{\left({U}_{1},{U}_{2}\right)}^{2}}{ct}\right)\parallel {f}_{1}{\parallel }_{{L}^{2}\left(X\right)}{\parallel {f}_{2}\parallel }_{{L}^{2}\left(X\right)},\phantom{\rule{1.em}{0ex}}\forall t>0,$

for every ${f}_{i}\in {L}^{2}\left(X\right)$ with supp${f}_{i}\subset {U}_{i}$, $i=1,2$.

Denote by ${H}_{L}^{p}\left(X\right)$ the Hardy space associated with $L$. The authors establish a Hörmander-type spectral multiplier theorem for $L$ on ${H}_{L}^{p}\left(X\right)$ for $0. Precisely, let $\varphi$ be a non-negative ${C}_{0}^{\infty }$ function on $ℝ$ such that

$supp\varphi \subseteq \left(\frac{1}{4},1\right)\phantom{\rule{1.em}{0ex}}\text{and}\phantom{\rule{1.em}{0ex}}\sum _{\ell \in ℤ}\varphi \left({2}^{-\ell }\lambda \right)=1\phantom{\rule{1.em}{0ex}}\forall \lambda >0·$

If $0 and the bounded measurable function $m:\left[0,\infty \right)\to ℂ$ satisfies

${C}_{\varphi ,s}=\underset{t>0}{sup}{\parallel \varphi \left(·\right)m\left(t·\right)\parallel }_{{C}^{s}}+|m\left(0\right)|<\infty$

for some $s>n\left(1/p-1/2\right)$, then $m\left(L\right)$ is bounded from ${H}_{L}^{p}\left(X\right)$ to ${H}_{L}^{p}\left(X\right)$. If ${C}_{\varphi ,s}<\infty$ for all $s>0$, then $m\left(L\right)$ is bounded on ${H}_{L}^{p}\left(X\right)$ for all $0.

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)