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

The heat semigroup {T(t)}t≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{T(t)\}_{t \ge 0}$$\end{document} defined on homogeneous Besov spaces B˙p,qs(Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{B}_{p,q}^s(\mathbb {R}^n)$$\end{document} is considered. We show the decay estimate of T(t)f∈B˙p,1s+σ(Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T(t)f \in \dot{B}_{p,1}^{s+\sigma }(\mathbb {R}^n)$$\end{document} for f∈B˙p,∞s(Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f \in \dot{B}_{p,\infty }^s(\mathbb {R}^n)$$\end{document} with an explicit bound depending only on the regularity index σ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma >0$$\end{document} 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(·)f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T(\cdot )f$$\end{document} in space and time. To refine the previous results, we show explicit pointwise estimates of higher order derivatives of the power function |ξ|σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\xi |^{\sigma }$$\end{document} for σ∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma \in \mathbb {R}$$\end{document}. In addition, we also refine the L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1$$\end{document}-estimate of the derivatives of the heat kernel.


Introduction
Consider the heat equation ∂ t u = u in (0, ∞) × R n with initial condition u(0) = f in R n , n ≥ 1.Then, the solution u is given by u(t) = T (t) f at least formally, where {T (t)} t≥0 is the heat semigroup defined by T (t) := F −1 e −t|ξ | 2 F for t ≥ 0. Here, F and F −1 denote the Fourier transform and inverse Fourier transform, respectively.
In this paper, we consider the function u(t) := T (t) f in the case where f belongs to the homogeneous Besov spaces Ḃs p,q (R n ) for 1 ≤ p, q ≤ ∞ and s ∈ R. Our purpose is to show the decay estimate of u(t) = T (t) f ∈ Ḃs+σ p,1 (R n ) for f ∈ Ḃs p,∞ (R n ) with an explicit bound depending only on the regularity index σ > 0 and space dimension n.In addition, we obtain an explicit lower bound of the radius of convergence of the Taylor expansion of u(•) = T (•) f in space and time.These results may be regarded as an improved version compared with [23].
To refine the decay estimate of {T (t)} t≥0 , we show explicit pointwise estimates of higher order derivatives of the power function |ξ | σ for σ ∈ R, where notice that |ξ | σ is the symbol of the fractional Laplacian (− ) σ/2 = F −1 |ξ | σ F. The pointwise estimates of |ξ | σ play an important role in giving an explicit bound appearing in the decay estimate of {T (t)} t≥0 .Moreover, we also refine the L 1 -estimate of the derivatives of the heat kernel in [23,Proposition 5.4].This improvement allows us to enlarge a lower bound estimate of the radius of convergence of the Taylor expansion.
Before stating our main results, let us define some function spaces.Take ϕ ∈ S (R n ) such that supp ϕ = {ξ ∈ R n | 1/2 ≤ |ξ | ≤ 2}, ϕ(ξ ) > 0 for 1/2 < |ξ | < 2, and j∈Z ϕ(2 − j ξ) = 1 for all ξ ∈ R n \ {0}, where S (R n ) denotes the Schwartz space.Then, for 1 ≤ p, q ≤ ∞ and s ∈ R, the homogeneous Besov spaces Ḃs p,q (R n ) are defined by where S (R n ) denotes the dual space of S (R n ), P(R n ) is the subspace of S (R n ) consisting of polynomials with complex coefficients, and {ϕ j } j∈Z is defined by ϕ j := F −1 [ϕ(2 − j ξ)] for j ∈ Z.In addition, we also define the Sobolev spaces Our first result is the decay estimate of solutions to the heat equation with an explicit bound.In what follows, let denote the gamma function and let [a] denote an integer such that [a] ≤ a < [a] + 1 for a ∈ R. We also set constants M σ,n and K n as follows: M σ,n := max (σ/2) σ/2 , (25/2) n/8 ( Theorem 1.1 Let 1 ≤ p ≤ ∞ and s ∈ R. For every f ∈ Ḃs p,∞ (R n ), it holds that T (t) f ∈ Ḃs+σ p,1 (R n ) for all σ > 0 and t > 0 with the estimate Remark 1.2 (i) The decay estimate of T (t) f ∈ Ḃs+σ p,1 (R n ) for f ∈ Ḃs p,∞ (R n ) was first shown by Kozono-Ogawa-Taniuchi [16,Lemma 2.2].However, it is unknown the dependence of a bound on n, p, s, and σ in [16,Lemma 2.2].Then, the second author [23,Lemma 5.1] refined the decay estimate by considering the dependence on the regularity index σ .In contrast to these results, we give an explicit bound depending only on σ and n.Moreover, we also refined the increasing rate of σ as compared with [23,Lemma 5.1].
(ii) Concerning the increasing rate of σ for fixed n, we have where In addition, since the Stirling formula yields , we obtain ∼ n 3n/8 (2πe 2 ) −3n/16 n 15/8 ϕ 3/4 By using the Stirling formula again, we have which implies that the increasing rate of n for fixed σ is given by where This condition is used for the estimate of the Fourier multipliers.In addition, the powers appearing in K n are related to 3/4, which comes from the interpolation inequality on Ḃs p,q (R n ).However, the value 3/4 is not essential in the estimate.
As a result of Theorem 1.1, we may show the two results on the analyticity of the bounded linear operator T (•) and the complex-valued function T (•) f , respectively.In addition, we also give an explicit lower bound of the radius of convergence of the Taylor expansion.We first state the result on the analyticity of T (•).In the following, In addition, it holds that Next, we shall state the result on the analyticity of T (•) f , namely, the space-time analyticity of solutions to the heat equation.Before doing so, we should mention the problem which the homogeneous Besov spaces Ḃs p,q (R n ) have.By recalling the definition of Ḃs p,q (R n ), we see that the function f ∈ Ḃs p,q (R n ) is regarded as an element of the quotient space S (R n )/P(R n ), so it does not make sense to consider the value of f (x) ∈ C at x ∈ R n .However, we may overcome this problem by introducing a suitable realization.More precisely, we make use of the result given by Bourdaud [2, Theorem 4.1 (1)].Then we have the following consequence: Then there exists a unique continuous linear operator : Ḃs p,q (R n ) → S (R n ) such that has the following properties: By regarding the element f ∈ S (R n ) as a function of Ḃs p,q (R n ), we may consider the value of f (x) ∈ C. Therefore, we introduce the function space Ḃs Then for every f ∈ Ḃs p,q (R n ), we may take g ∈ Ḃs p,q (R n ) such that f = g, so we may regard f Ḃs p,q (R n ) := g Ḃs p,q (R n ) .By using the notation Ḃs p,q (R n ), we may state our main result on the space-time analyticity of solutions to the heat equation.

Remark 1.7 (i)
In this paper, we use the definition of the homogeneous Besov spaces which is based on the sequence {ϕ j } j∈Z .On the other hand, we may define the homogeneous Besov spaces in another way instead of using {ϕ j } j∈Z ; By taking a suitable non-negative function ϕ ∈ C ∞ 0 (R n \ {0}) and setting ϕ λ := F −1 [ϕ(λξ )] for λ > 0, we may introduce the norm , which is equivalent to that of our definition.Moreover, this norm has the property such that f (λ•) Ḃs p,q (R n ) = λ s−n/ p f Ḃs p,q (R n ) for all λ > 0, so we may expect that the computations in this paper will become simpler.However, since our results focus on giving an explicit bound, we have to pay attention to using another definition of the function spaces, i.e., another norm.In particular, to compare with the previous result of the second author [23], we use the same definition as in [23].
(ii) For recent results on the analyticity of solutions to the heat equation, Dong-Zhang [5] and Zhang [25] showed the time analyticity in the case of complete noncompact Riemannian manifolds M. Han-Hua-Wang [8] also considered the case of weighted graphs G.In addition, Zeng [24] showed the time analyticity of solutions to the biharmonic heat equation and the heat equation with potentials in the case of M. Concerning the nonlinear problems, there are recent contributions on the analyticity of solutions to the Navier-Stokes system [1,4,17], Schrödinger equation [9-11, 19, 20], Burgers equation [12,13], and primitive equations [6].This paper is organized as follows: We give some notations and key estimates in the next section.In Sect.3, we show the explicit estimates of the differential operators (− ) σ/2 and ∂ α x on Ḃs p,q (R n ).Moreover, we prove Theorem 1.1, i.e., the decay estimate of ) with an explicit bound depending only on the regularity index σ > 0 and space dimension n.Section 4 is devoted to the estimate of a lower bound of the radius of convergence (1.2) of the Taylor expansion, namely, the proof of Theorems 1.3 and 1.5.In Appendix, we show explicit pointwise estimates of higher order derivatives of the power function |ξ | σ for σ ∈ R.

Preliminaries
In this section, let us give some notations used throughout this paper.Let denotes a binomial coefficient.We also set , where is the gamma function, i.e., (a) := ∞ 0 λ a−1 e −λ dλ with a > 0. Concerning the Fourier transform F and its inverse F −1 , we use the following definitions It is well-known that F and F −1 can be extended to the operators from [22,Theorem 2.25], we see that (− ) σ/2 can be extended to the operator from In what follows, the family {T (t)} t≥0 of the operators denotes the heat semigroup, i.e., T (t Next, we state explicit pointwise estimates of higher order derivatives of the power The proof of Proposition 2.1 is given in Appendix.By using Proposition 2.1, we may show the following estimates, which play an important role in giving an explicit bound in Theorem 1.1: Hence we obtain (2.1).Concerning the estimate (2.2), assume that γ ≤ α.Then we see that This completes the proof of Proposition 2.2.
Finally, we give the L 1 -estimate of the derivatives of the heat kernel G t (x) := (4π t) −n/2 e −|x| 2 /(4t) for (t, x) ∈ (0, ∞) × R n .The following proposition is regarded as an improved estimate in [23,Proposition 5.4]: in the case n ≥ 2. Thus we see by In addition, the substitution r = (4t) 1/2 λ yields for all t > 0.Here we notice that the above identity is still valid in the case n = 1.Therefore, since G t+τ = G t * G τ for all t, τ > 0, the Young convolution inequality implies that for all N ∈ N and t > 0. This proves Proposition 2.3.

Explicit Estimates of Some Operators on the Homogeneous Besov Spaces
In this section, we shall show the explicit estimates of differential operators (− ) σ/2 and ∂ α x on Ḃs p,q .We also prove the refined decay estimate of {T (t)} t≥0 , namely, Theorem 1.1.

Estimates of Differential Operators
Let us set and show the following theorem: To show Theorem 3.1, we consider the symbols |ξ | σ and (iξ) α of the Fourier multiplier operators We begin with giving the L 1 -estimate of the function F −1 f , where the estimate will be used to obtain the estimate of the Fourier multipliers.
Proof Let R > 0 be arbitrary.Then, by the Cauchy-Schwarz inequality and Parseval identity In addition, since m ∈ N satisfies 2m ≥ n + 1, it holds by the Cauchy-Schwarz inequality and Hence, by setting , we observe that which completes the proof of Proposition 3.2.
Proof Notice that we have for all β ∈ N n 0 satisfying |β| = m, we see by the Leibniz rule and Proposition 2.2 that Hence it holds by (3.2) that Here we note that which implies the desired estimate.In the case ρ(ξ ) = (iξ) α , for all β ∈ N n 0 satisfying |β| = m, we also see by the Leibniz rule and Proposition 2.2 that Thus we have the desired estimate in the same way.This completes the proof of Proposition 3.3.
By combining the estimates in Propositions 3.2 and 3.3, we may show the following estimate, which provides the proof of Theorem 3.1 immediately.Here we set Then it holds that Proof We see by Proposition 3.2 that for all k ∈ Z, provided 2m ≥ n + 1.We also notice that which implies the desired estimate.In the case ρ(ξ ) = (iξ) α , we may show the desired estimate in a similar manner.This proves Proposition 3.4.Corollary 3.5 Let σ ∈ R and α ∈ N n 0 .Then it holds that for all k ∈ Z, where M σ,n and M α,n are the same constants defined by (3.1).
Proof of Theorem 3.1 First we notice that g = k∈Z (ϕ k * g) for all g ∈ L p and ϕ j * ϕ k = 0 for all j, k ∈ Z such that | j − k| ≥ 2 from the definition of {ϕ j } j∈Z .Hence, we see by the Young convolution inequality that for all k ∈ Z from simple substitutions, by applying Corollary 3.5 and using j+1 k= j−1 for all j ∈ Z. Thus we have the desired estimates.This completes the proof of Theorem 3.1.

Decay Estimate of the Heat Semigroup
In this subsection, we shall show the refined decay estimate of T (t) f ∈ Ḃs+σ p,1 for f ∈ Ḃs p,∞ , i.e., Theorem 1.1.To obtain an explicit bound, we make use of the L 1estimate of the derivatives of the heat kernel given in Proposition 2.3.We begin with showing the following lemma: Lemma 3.6 Let 1 ≤ p, q ≤ ∞ and s ∈ R. For every f ∈ Ḃs p,q , it holds that T (t) f ∈ Ḃs+2N p,q for all N ∈ N and t > 0 with the estimate , where K n is the same constant defined by (3.1).
Proof Notice that T (t) f = G t * f holds for all t > 0, where G t (x) := (4π t) −n/2 e −|x| 2 /(4t) for (t, x) ∈ (0, ∞) × R n .Therefore, in a similar manner to the proof of Theorem 3.1, we have ϕ j * (T (t) f ) L p ≤ ϕ j * f L p j+1 k= j−1 ϕ k * G t L 1 for all j ∈ Z and t > 0. In addition, by the Young convolution inequality, we observe that for all N ∈ N, k ∈ Z, and t > 0. Since it holds by Corollary 3.5 that by applying Proposition 2.3 and using for all N ∈ N, j ∈ Z, and t > 0. Thus we have the desired estimate.This proves Lemma 3.6.
Notice that the decay estimate in Lemma 3.6 is valid in the case N ∈ N.Then, we shall extend the decay estimate to the general regularity index σ > 0. To this end, we show the interpolation inequality of Ḃs p,q with an explicit bound.By combining Lemma 3.6 and the interpolation inequality, we may show Theorem 1.1. , Hence, by taking j * ∈ Z so that .

Radius of Convergence of the Taylor Expansion
In this section, we give the proof of Theorems 1.3 and 1.5.Namely, we calculate a lower bound of the radius (1.2) of convergence of the Taylor expansion appearing in Theorems 1.3 and 1.5.To this end, we make use of Theorems 1.1 and 3.1.We also notice that the following continuous embedding Ḃn/p p,1 ⊂ L ∞ , which yields the pointwise estimate of the remainder term of the Taylor expansion (1.3).We first give the proof of Theorem 1.5.Theorem 1.3 may be shown in a similar manner.
Proof of Theorem 1. 5 We focus on a derivation of a lower bound (1.2) of the radius of convergence of the Taylor expansion (1.3).Assume that s − n/ p / ∈ N and ∈ Ḃs p,∞ .Set u(t, x) := (T (t) f )(x) for (t, x) ∈ (0, ∞) × R n and consider the remainder term of the Lagrange form of the Taylor expansion (1.3) at an arbitrary ) with some 0 < θ < 1.Since we consider the case where k * + j * is sufficiently large, we may take k 0 , j 0 ∈ N 0 so that k 0 ≤ k * , j 0 ≤ j * , and k 0 + 2 j 0 > s − n/ p.Hence, by setting k := k * − k 0 and j := j * − j 0 , we observe that Here, since Ḃn/p p,1 ⊂ L ∞ and since ∂ )), we see by Theorem 3.1 that for all α ∈ N n 0 satisfying |α| = k, where C > 0 is a constant depending on k 0 and j 0 but not on k * and j * .In addition, noting that σ 0 := k 0 + 2 j 0 + n/ p − s > 0 and x j u(t * ), by combining Theorems 1.1 and 3.1, we deduce that Concerning the constants depending on α and j, we have where C > 0 is a constant independent of α and j.In addition, it holds that provided that |α|+ j is sufficiently large.Therefore, noting that |α|=k 1/α! = n k /k!, we obtain , where P(k, j) := (k + j) σ 0 /2+3n/8 (k +1) n/2 ( j +1) n/2 .In addition, since (a+b) a+b ≤ 2 a+b a a b b for a, b ≥ 0 from [23,Proposition 5.3] and since (1 + a/b) b ≤ e a for a, b > 0, we have Hence we see by Here we recall that k 0 , j 0 ∈ N 0 are the fixed indices and k = k * − k 0 and j = j * − j 0 .Notice that for any x, x 0 ∈ R n and t * > 0, by taking k * ∈ N sufficiently large, we have Hence, the remainder term of the Taylor expansion (1.3) converges to 0, provided that |t − t 0 | is sufficiently small so that holds.Once we see that (1.3) is valid for t > 0 satisfying the above condition, in the same way as above, we may show that (1.3) holds, provided (1.2).This completes the proof of Theorem 1.5.
Proof of Theorem 1. 3 We see that for all j ∈ N 0 and σ > 0 by virtue of Theorems 1.1 and 3.1, which yields T (t) ∈ L( Ḃs p,∞ , Ḃs+σ p,1 ) with the estimate Therefore, in a similar manner to the proof of Theorem 1.5, we may show the desired result.This completes the proof of Theorem 1.3.

Proposition A.2 It holds that
it holds that