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

Abstract

AbstractThe heat semigroup $$\{T(t)\}_{t \ge 0}$$ { T ( t ) } t ≥ 0 defined on homogeneous Besov spaces $$\dot{B}_{p,q}^s(\mathbb {R}^n)$$ B ˙ p , q s ( R n ) is considered. We show the decay estimate of $$T(t)f \in \dot{B}_{p,1}^{s+\sigma }(\mathbb {R}^n)$$ T ( t ) f ∈ B ˙ p , 1 s + σ ( R n ) for $$f \in \dot{B}_{p,\infty }^s(\mathbb {R}^n)$$ f ∈ B ˙ p , ∞ s ( R n ) with an explicit bound depending only on the regularity index $$\sigma >0$$ σ > 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 $$T(\cdot )f$$ T ( · ) f in space and time. To refine the previous results, we show explicit pointwise estimates of higher order derivatives of the power function $$|\xi |^{\sigma }$$ | ξ | σ for $$\sigma \in \mathbb {R}$$ σ ∈ R . In addition, we also refine the $$L^1$$ L 1 -estimate of the derivatives of the heat kernel.