Remarks on the smoothing effect of the heat semigroup on [formula omitted] ([formula omitted])

Abstract

We consider the heat semigroup etΔ:Ḃp,∞s(Rn)→Ḃp,∞s(Rn) for 1≤p≤∞ and s∈R. We show that etΔf∈BC((0,∞);Ḃp,∞s(Rn)) for all f∈Ḃp,∞s(Rn). In addition, we reveal a sufficient condition of f∈Ḃp,∞s(Rn) for ensuring the C0-property etΔf∈BC([0,∞);Ḃp,∞s(Rn)). We also give examples of f∈Ḃp,∞s(Rn) which have the property etΔf∉BUC((0,1);Ḃp,∞s(Rn)).