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

Let H = − Δ + | x | 2 be the harmonic oscillator on R n . 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 H α , 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 H α in the sense that similar estimates might fail with the classical Besov spaces.