On some PDEs involving homogeneous operators. Spectral analysis, semigroups and Hardy inequalities

Abstract

Motivated by the analysis of homogeneous operators and semigroups in [10], in this paper we perform an spectral analysis of homogeneous operators and give simple characterization of generators of homogeneous semigroups and of homogeneous sectorial operators. We also analyse homogeneous perturbations of homogeneous operators and semigroups. We use these results to study some parabolic PDEs involving homogeneous operators including some fractional diffusion problems. We show a deep connection of these results with some Gagliardo–Nirenberg and Hardy type inequalities.