Spectral upper bound for the torsion function of symmetric stable processes

Abstract

We prove a spectral upper bound for the torsion function of symmetric stable processes that holds for convex domains in R d \mathbb {R}^d . Our bound is explicit and captures the correct order of growth in d d , improving upon the existing results of Giorgi and Smits [Indiana Univ. Math. J. 59 (2010), pp. 987–1011] and Biswas and Lőrinczi [J. Differential Equations 267 (2019), pp. 267–306]. Along the way, we make progress towards a torsion analogue of Chen and Song’s [J. Funct. Anal. 226 (2005), pp. 90–113] two-sided eigenvalue estimates for subordinate Brownian motion.