Regularity estimates for the Cauchy problem to a parabolic equation associated to fractional harmonic oscillators

Abstract

Let H=−Δ+|x|2 be the harmonic oscillator on Rn. In this paper, we prove estimates on Besov spaces associated to the operator H and the end-point maximal regularity estimates for the fractional harmonic oscillator Hα, 0<α≤1, on the Besov spaces associated to the harmonic oscillator H. These spaces are the appropriate function spaces for the study of estimates on Besov type spaces and the end-point maximal regularity estimates for the fractional power Hα in the sense that similar estimates might fail with the classical Besov spaces.