×

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
PDF BibTeX XML Cite
Full Text: DOI