\(L^p\) - \(L^q\) Norms of Spectral Multipliers

This presentation is based on the joint work with M. Ruzhansky, which has recently been published in Rottensteiner and Ruzhansky (Arch. Math. (Basel) 120:507–520, 2023). We present an update on the multiplier theorems by Akylzhanov and Ruzhansky (J. Funct. Anal. 278(3):108324, 2020) by extending relevant \(L^p\) - \(L^q\) norm estimates to spectral multipliers of left-invariant weighted subcoercive operators on unimodular Lie groups. In particular, this includes spectral multipliers of Laplacians, sub-Laplacians and Rockland operators. As an application, we obtain, e.g., time asymptotics for the \(L^p\) - \(L^q\) norms of the heat kernels and Sobolev-type embeddings.