Space–time analytic smoothing effect of the heat semigroup defined on homogeneous Besov spaces

Abstract

We refine the decay estimate of the heat semigroup {T(t)}t≥0defined on homogeneous Besov spaces Ḃp,qs(Rn)for s∈R,p,q∈[1,∞], which is obtained by Kozono et al. (2003). In particular, we give an explicit representation of a constant appeared in the decay estimate of {T(t)}t≥0, which provides a space–time analytic smoothing effect of {T(t)}t≥0. As a by-product, we obtain a radius of convergence of the Taylor expansion exactly. Furthermore, it is also showed that {T(t)}t≥0is a bounded analytic C0-semigroup on Ḃp,qs(Rn)for s∈R,p,q∈[1,∞), where {T(t)}t≥0can be extended as an analytic function of t on the sector {t∈ℂ∖{0}||argt|<θ}with an explicitly given constant θ.

Keywords

• Heat semigroup
• Analyticity
• Homogeneous Besov spaces
• Fourier multiplier