Bounded $H^\infty$-calculus for vectorial-valued operators with Gaussian kernel estimates

TL;DR

This paper establishes that certain vector-valued operators with Gaussian kernel estimates have bounded $H^$-calculus across all $L^p$ spaces, extending known results from $L^2$ to a broader range.

Contribution

It proves bounded $H^$-calculus for vector-valued generators with Gaussian estimates in all $L^p$ spaces, generalizing previous $L^2$-based results.

Findings

Bounded $H^$-calculus extends from $L^2$ to all $L^p$ spaces for the operators considered.

Application to elliptic operators with matrix-valued potentials in $L^1_{loc}$.

Operators with Gaussian kernel estimates satisfy bounded $H^$-calculus across $p  (1,0)$.

Abstract

We prove that the vector-valued generator of a bounded holomorphic semigroup represented by a kernel satisfying Gaussian estimates with bounded $H^\infty$-calculus in $L^2(\mathbb R^d;\mathbb C^m)$ admits bounded $H^\infty$-calculus for every $p\in (1,\infty)$. We apply this result to the elliptic operator $-{\rm div}(Q\nabla)+V$, where the potential term V is a matrix-valued function whose entries belong to $L^1_{\rm loc}(\mathbb R^d)$ and, for almost every $x\in \mathbb R^d$, $V(x)$ is a symmetric and nonnegative definite matrix.