## Journal of the Mathematical Society of Japan

### Weighted inequalities for holomorphic functional calculi of operators with heat kernel bounds

#### Abstract

Let $\mathscr{X}$ be a space of homogeneous type. 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 $\mathscr{X}$. For appropriate bounded holomorphic functions $b$, we can define the operators $b(L)$ on $L^p({\Omega})$, $1\leq p\leq \infty$. We establish conditions on positive weight functions $u, v$ such that for each $p$, $1, there exists a constant $c_p$ such that $\int_{\Omega} |b(L)f(x)|^p u(x)d\mu(x) \leq c_p ||b||^p_{\infty} \int_{\Omega} |f(x)|^p v(x)d\mu(x)$for all $f\in L^p(vd\mu)$.

Applications include two-weight $L^p$ inequalities for Schrödinger operators with non-negative potentials on $\bm{R}^n$ and divergence form operators on irregular domains of $\bm{R}^n$.

#### Article information

Source
J. Math. Soc. Japan, Volume 57, Number 4 (2005), 1129-1152.

Dates
First available in Project Euclid: 14 June 2006

https://projecteuclid.org/euclid.jmsj/1150287306

Digital Object Identifier
doi:10.2969/jmsj/1150287306

Mathematical Reviews number (MathSciNet)
MR2183586

Zentralblatt MATH identifier
1084.42010

#### Citation

DUONG, Xuan Thinh; YAN, Lixin. Weighted inequalities for holomorphic functional calculi of operators with heat kernel bounds. J. Math. Soc. Japan 57 (2005), no. 4, 1129--1152. doi:10.2969/jmsj/1150287306. https://projecteuclid.org/euclid.jmsj/1150287306

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