Maximal Hörmander Functional Calculus on L p Spaces and UMD Lattices

Abstract

Abstract Let $A$ be a generator of an analytic semigroup having a Hörmander 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 Hörmander spectral multipliers decaying sufficiently fast at $\infty $, there holds a maximal estimate $\| \sup _{t> 0} |m(tA)f|\, \|_{L^p(\Omega ,Y)} \lesssim \|f\|_{L^p(\Omega ,Y)}$. We also show square function estimates $\left \| \left ( \sum _k \sup _{t> 0} |m_k(tA)f_k|^2 \right )^{\frac 12} \right \|_{L^p(\Omega ,Y)} \lesssim \left \| \left ( \sum _k |f_k|^2 \right )^{\frac 12} \right \|_{L^p(\Omega ,Y)}$ for suitable families of spectral multipliers $m_k$, which are even new for the euclidean Laplacian on scalar valued $L^p(\ensuremath {{\mathbb {R}}}^d)$. 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 Hörmander functional calculus on some $L^p(\Omega ,Y)$ and discuss examples of lattices $Y$ and non-self-adjoint operators $A$ fitting our context.