Maximal Hörmander Functional Calculus on Lp Spaces and UMD Lattices (2203.03263v1)

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.