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

Abstract

Let $𝒳$ be a space of homogeneous type. Assume that $L$ has a bounded holomorphic functional calculus on $L^2\left(\mathrm{\Omega }\right)$ and $L$ generates a semigroup with suitable upper bounds on its heat kernels where $\mathrm{\Omega }$ is a measurable subset of $𝒳$. For appropriate bounded holomorphic functions $b$, we can define the operators $b\left(L\right)$ on $L^p\left(\mathrm{\Omega }\right)$, $1\le p\le \mathrm{\infty }$. We establish conditions on positive weight functions $u,v$ such that for each $p$, , there exists a constant $c_p$ such that $${\int }_{\mathrm{\Omega }}\left|b\right(L\left)f\right(x\left){|}^{p}u\right(x\left)d\mu \right(x)\le c_p|\left|b\right|{|}_{\mathrm{\infty }}^p{\int }_{\mathrm{\Omega }}\left|f\right(x\left){|}^{p}v\right(x\left)d\mu \right(x)$$for all $f\in L^p\left(vd\mu \right)$.

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