Refined Decay Estimate and Analyticity of Solutions to the Linear Heat Equation in Homogeneous Besov Spaces

Abstract

The heat semigroup {T(t)}_{t ge 0} defined on homogeneous Besov spaces dot{B}_{p,q}^s(mathbb {R}^n) is considered. We show the decay estimate of T(t)f in dot{B}_{p,1}^{s+sigma }(mathbb {R}^n) for f in dot{B}_{p,infty }^s(mathbb {R}^n) with an explicit bound depending only on the regularity index sigma >0 and space dimension n. It may be regarded as a refined result compared with that of the second author (Takeuchi in Partial Differ Equ Appl Math 4:100174, 2021). As a result of the refined decay estimate, we also improve a lower bound estimate of the radius of convergence of the Taylor expansion of T(cdot )f in space and time. To refine the previous results, we show explicit pointwise estimates of higher order derivatives of the power function |xi |^{sigma } for sigma in mathbb {R}. In addition, we also refine the L^1-estimate of the derivatives of the heat kernel.