Maximal H\"ormander Functional Calculus on Lp Spaces and UMD Lattices

TL;DR

This paper establishes maximal and square function estimates for H{"o}rmander spectral multipliers on $L^p$ spaces with UMD lattice targets, extending functional calculus results to a broad class of operators.

Contribution

It introduces new maximal and square function estimates for spectral multipliers on $L^p$ spaces with UMD lattices, including cases for non-self-adjoint operators.

Findings

Maximal estimates for spectral multipliers with decay conditions.

Square function estimates for families of spectral multipliers.

Applications to wave propagators and Bochner--Riesz means.

Abstract

Let $A$ be a generator of an analytic semigroup having a H{\"o}rmander functional calculus on $X = L^p(\Omega ,Y)$, where $Y$ is a UMD lattice. Using methods from Banach space geometry in connection with functional calculus, we show that for H{\"o}rmander spectral multipliers decaying sufficiently fast at $\infty$, there holds a maximal estimate $\| \sup_{t \geq 0} |m(tA)f|\, \|_{L^p(\Omega ,Y)} \lesssim \|f\|_{L^p(\Omega ,Y)}$. We also show square function estimates $\left\| \left( \sum_k \sup _{t \geq 0} |m_k(tA)f_k|^2 \right)^{\frac12} \right\|_{L^p(\Omega ,Y)} \lesssim \left\| \left( \sum _k |f_k|^2 \right)^{\frac12} \right\|_{L^p(\Omega ,Y)}$ for suitable families of spectral multipliers $m_k$, which are even new for the euclidean Laplacian on scalar valued $L^p(\mathbb{R}^d)$. As corollaries, we obtain maximal estimates for wave propagators and Bochner--Riesz means. Finally, we illustrate the results by giving several examples of operators $A$ that admit a H{\"o}rmander functional calculus on some $L^p(\Omega ,Y)$ and discuss examples of lattices $Y$ and non-self-adjoint operators $A$ fitting our context.