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 \(\mathbb R^n\).


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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