## Weighted inequalities of holomorphic functional calculi of operators with heat kernel bounds.(English)Zbl 1084.42010

Let $$(X,d,\mu)$$ be a space of homogeneous type in the sense of Coifman and Weiss. Moreover, if $$X$$ has infinite measure, assume that all annuli in $$X$$ are not empty. Assume that $$L$$ has a bounded holomorphic functional calculus on $$L^2(\Omega)$$ and $$L$$ generates a semigroup with suitable upper bounds on its heat kernels where $$\Omega$$ is a measurable subset of $$X$$. For appropriate bounded holomorphic functions $$b$$, the operator $$b(L)$$ is well-defined on $$L^p(\Omega)$$ for $$1<p<\infty$$. Let $$1/p+1/p'=1$$. Define $D_p=\Big\{0\leq w<\infty\;\mu\text{-a.\,e.:}\quad \int_\Omega {w(x)^{1-p'}\over(1+\mu(B(x_0,d(x_0,x))))^{p'}}\, d\mu(x)<\infty\Big\}$ and $Z_p=\Big\{0\leq w<\infty\;\mu\text{-a.\,e.:}\quad \int_\Omega {w(x)\over(1+\mu(B(x_0,d(x_0,x))))^p}\, d\mu(x)<\infty\Big\}.$ It is easy to check that $$D_p$$ and $$Z_p$$ are independent of $$x_0$$.
The main result of this paper is to prove that for appropriate operators $$L$$ and appropriate bounded holomorphic functions $$b$$, if $$u\in Z_p$$ (resp. $$v\in D_p$$), then there exists a weight $$0<v<\infty$$ $$\mu$$-a. e. (resp. $$0<u<\infty$$ $$\mu$$-a. e.) such that for all $$f\in L^p(v\,d\mu)$$, $\int_\Omega| b(L)f(x)| ^pu(x)\,d\mu(x)\leq C_p\| b\| ^p_{\infty} \int_\Omega| f(x)| ^pv(x)d\mu(x).$ Moreover, for $$0<\alpha<1$$, $$v$$ (resp. $$u$$) can be chosen such that $$v^\alpha\in Z_p$$ (resp. $$u^\alpha\in D_p$$).
Finally, the authors also give some applications to two-weighted $$L^p$$ inequalities for Schrödinger operators with non-negative potentials on $$\mathbb R^n$$ and divergence form operators on irregular domains of $$\mathbb R^n$$.

### MSC:

 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25 Maximal functions, Littlewood-Paley theory 47A30 Norms (inequalities, more than one norm, etc.) of linear operators 47A60 Functional calculus for linear operators 43A85 Harmonic analysis on homogeneous spaces 35J10 Schrödinger operator, Schrödinger equation
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