Maximal Hörmander Functional Calculus on Lp Spaces and UMD Lattices

Abstract

Let $A$ be a generator of an analytic semigroup having a H{ö}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{ö}rmander spectral multipliers decaying sufficiently fast at $\infty$, there holds a maximal estimate $\Vert {\sup }_{t\ge 0}|m(tA)f|{\Vert }_{L^p(\Omega ,Y)}\lesssim \Vert f{\Vert }_{L^p(\Omega ,Y)}$. We also show square function estimates ${\Vert {\left({\sum }_k{\sup }_{t\ge 0}|m_k(tA)f_k{|}^2\right)}^{\frac{1}{2}}\Vert }_{L^p(\Omega ,Y)}\lesssim {\Vert {\left({\sum }_k|f_k{|}^2\right)}^{\frac{1}{2}}\Vert }_{L^p(\Omega ,Y)}$ 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{ö}rmander functional calculus on some $L^p(\Omega ,Y)$ and discuss examples of lattices $Y$ and non-self-adjoint operators $A$ fitting our context.

International Mathematics Research Notices, Oxford University Press (OUP), 2022

References in corpus (3)

• Two-sided estimates of heat kernels on metric measure spaces
• $L^p$-spectral multipliers for some elliptic systems
• Vector valued $q$-variation for differential operators and semigroups II