Higher-order expansion of solutions for a damped wave equation

Abstract

We study the Cauchy problem for a damped wave equation. Our main result is the $(N + 1)$ th order approximation of solutions by the solution of the corresponding parabolic equation for arbitrary $N \in N$ in the $L^{p}$ framework. This means that our results can be applied to nonlinear damped wave equations.

Keywords

damped wave equation asymptotic profile the Cauchy problem higher-order expansion of solution

1. Introduction

In this paper we consider the Cauchy problem for a damped wave equation $\{\begin{matrix} \partial_{t}^{2} u - Δ u + \partial_{t} u = 0, & t > 0, x \in R^{n}, \\ u (0, x) = g_{0} (x), \partial_{t} u (0, x) = g_{1} (x), & x \in R^{n}, \end{matrix}$ (1.1) where $g_{0}$ and $g_{1}$ are given functions.

Our purpose is to study the large time behavior of the solution of (1.1). More precisely we establish the approximation of the solution u by the solution of the corresponding parabolic equation $\{\begin{matrix} \partial_{t} v - Δ v = 0, & t > 0, x \in R^{n}, \\ v (0, x) = g_{0} (x) + g_{1} (x), & x \in R^{n} \end{matrix}$ (1.2) in the $L^{p}$ framework. We use the Fourier multiplier representation of the solution and apply the Fourier multiplier theorem. As a corollary, we obtain the approximation of the solution u by the Gauss kernel $G_{t} (x) = {(\frac{1}{4 π t})}^{n / 2} e^{- | x |^{2} / (4 t)}$ and we observe the optimality of our results with respect to the order of the decay rate.

Let us describe some previous results on decay estimates and the asymptotic behavior of the solutions for the damped wave equation (1.1). In a pioneering work in this direction, Matsumura [24] obtained classical decay estimates in the $L^{2}$ base spaces. After that, it has been asserted that the damped wave equation has a diffusive structure as $t \to \infty$ ([2,21] and [5]). This means that the solution of the damped wave equation is approximated by the Gauss kernel when t is large. Subsequently, many authors showed the diffusion phenomena of the solution of (1.1). The more general case is also well investigated (including nonlinear perturbations [8,10,11,13–15,18–28,30–34] and see also references therein). The previous works show that, up to the second order, the asymptotic behavior of the solution of (1.1) is same as that of solutions to (1.2). The main goal of this paper is a description of the asymptotic profiles of solutions to (1.1) for $t \to \infty$ . Regarding the above results for (1.1) and (1.2), one may expect that the asymptotic behavior of the solutions of (1.1) is as same as that of (1.2) for arbitrary order. However from the third-order expansion by the Gauss kernel, the asymptotic behavior of the solution of (1.1) is different from that of the solution of (1.2). Formally speaking, this implies that the effect of $\partial_{t}^{2} u$ in (1.1) is not negligible for the third-order expansion. In fact, Gallay and Raugel [8] considered the variable coefficient case with nonlinear perturbations for $n = 1$ and they suggested that this property does not hold for higher-order approximation [8, Remark 4, p. 51]. Furthermore, Orive, Zuazua and Pazoto [29] considered the variable coefficient case and they proved the higher-order expansion of the arbitrary order by the Bloch wave decomposition in $L^{2} \times H^{- 1}$ for all $n ⩾ 1$ . In contrast, their results seem to be difficult to simply apply to the arbitrary-order expansion of the solution of the nonlinear problem. Thus we only consider the constant coefficient case and we obtain the sharp higher-order asymptotic expansion in the $L^{p}$ framework under the minimal regularity condition for the initial data. Especially, our results can deal with the asymptotic behavior of solution of (1.1) in $L^{p}$ , $1 ⩽ p ⩽ \infty$ . One can easily guess that our results are useful to show the higher-order expansion of the solution to the nonlinear problem.

To state our main results, we introduce some notation. We denote by $[K]$ the integer satisfying $K - 1 < [K] ⩽ K$ , e.g. for $N \in Z_{+} : = N \cup {0}$ , $[\frac{N}{2}] : = \{\begin{matrix} \frac{N}{2} & if N is even, \\ \frac{N - 1}{2} & if N is odd . \end{matrix}$ (1.3) We denote the weighted $L^{1}$ space of Nth order and the corresponding weighted Sobolev space by $\begin{array}{rcl} L_{N}^{1} = L_{N}^{1} (R^{n}) : = {f \in L^{1} (R^{n}); (1 + | x |^{N}) f \in L^{1} (R^{n})}, \\ W_{N}^{[n / 2], 1} = W_{N}^{[n / 2], 1} (R^{n}) : = {f \in W^{[n / 2], 1} (R^{n}); (1 + | x |^{N}) f \in W^{[n / 2], 1} (R^{n})}, \end{array}$ where we identify $W_{N}^{0, 1} = L_{N}^{1}$ . For j, k and $ℓ \in Z_{+}$ , we put $ϕ_{j} (r) : = {(\frac{1}{1 / 2 + \sqrt{1 / 4 - r}})}^{2 j}, ψ (r) : = \frac{1}{2 \sqrt{1 / 4 - r}}$ (1.4) and $α_{j, k} : = {\frac{1}{j! k!} \frac{d^{k}}{d r^{k}} ϕ_{j} (r) |}_{r = 0}, β_{ℓ} : = {\frac{1}{ℓ!} \frac{d^{ℓ}}{d r^{ℓ}} ψ (r) |}_{r = 0} .$ (1.5) In particular, the norm ${∥ \cdot ∥}_{L^{p} (R^{n})} = {∥ \cdot ∥}_{L^{p}}$ is denoted by ${∥ \cdot ∥}_{p}$ . We also put $μ_{0 γ} : = \frac{1}{γ!} \int_{R^{n}} {(- y)}^{γ} g_{0} (y) d y, μ_{1 γ} : = \frac{1}{γ!} \int_{R^{n}} {(- y)}^{γ} g_{1} (y) d y$ (1.6) for $γ \in Z_{+}^{n}$ , which appear as the coefficients of the asymptotic expansion of the solution of (1.1) by the Gauss kernel depending on the initial data.

We can now formulate our main results.

Theorem 1.1.

Let $n = 1, 2, 3$ , $1 ⩽ r ⩽ q ⩽ \infty$ and $N \in Z_{+}$ . Assume that $(g_{0}, g_{1}) \in W^{[n / 2], r} \cap W^{[n / 2], q} \times L^{r} \cap L^{q}$ , then there exists a constant $C > 0$ such that the unique solution of (1.1) $u (t)$ satisfies the estimate $\begin{array}{rclr} ∥ u (t) - \frac{1}{2} \sum_{j = 0}^{[N / 2]} \sum_{k = 0}^{[N / 2] - j} α_{j, k} {(- t)}^{j} {(- Δ)}^{2 j + k} e^{t Δ} g_{0} \\ {- \sum_{j = 0}^{[N / 2]} \sum_{k = 0}^{[N / 2] - j} \sum_{ℓ = 0}^{[N / 2] - j - k} α_{j, k} β_{ℓ} {(- t)}^{j} {(- Δ)}^{2 j + k + ℓ} e^{t Δ} (\frac{1}{2} g_{0} + g_{1}) ∥}_{q} \\ ⩽ \{\begin{matrix} C t^{- (n / 2) (1 / r - 1 / q) - ([N / 2] + 1)} ({∥ g_{0} ∥}_{r} + {∥ g_{1} ∥}_{r}) + C e^{- δ t} ({∥ g_{0} ∥}_{W^{[n / 2], q}} + {∥ g_{1} ∥}_{q}), & t ⩾ 1, \\ C t^{- (n / 2) (1 / r - 1 / q) - [N / 2]} ({∥ g_{0} ∥}_{r} + {∥ g_{1} ∥}_{r}) + C e^{- δ t} ({∥ g ∥}_{W^{[n / 2], q}} + {∥ g_{1} ∥}_{q}), & 0 < t ⩽ 1, \end{matrix} & (1.7) \end{array}$ where $α_{j, k}$ , $β_{ℓ}$ are defined by (1.5) and $[\frac{N}{2}]$ is defined by (1.3) .

Combining Theorem 1.1 and the weighted $L^{1}$ estimate (1.16), we have the approximation of the solution by the Gauss kernel. This implies the lower bound of the estimate (1.7).

Corollary 1.2.

Let $n = 1, 2, 3$ , $1 ⩽ q ⩽ \infty$ and $N \in Z_{+}$ . Assume that $(g_{0}, g_{1}) \in W_{N}^{[n / 2], 1} \cap W^{[n / 2], \infty} \times L_{N}^{1} \cap L^{\infty}$ , then the unique solution of (1.1) $u (t)$ is in the class $C ([0, \infty); L_{N}^{1} \cap L^{\infty})$ and satisfies the estimate $\begin{array}{rclr} ∥ u (t) - \frac{1}{2} \sum_{j = 0}^{[N / 2]} \sum_{k = 0}^{[N / 2] - j} \sum_{| γ | ⩽ N - 2 (j + k)} α_{j, k} μ_{0 γ} {(- t)}^{j} {(- Δ)}^{2 j + k} \partial_{x}^{γ} G_{t} \\ {- \sum_{j = 0}^{[N / 2]} \sum_{k = 0}^{[N / 2] - j} \sum_{ℓ = 0}^{[N / 2] - j - k} \sum_{| γ | ⩽ N - 2 (j + k + ℓ)} α_{j, k} β_{ℓ} (\frac{1}{2} μ_{0 γ} + μ_{1 γ}) {(- t)}^{j} {(- Δ)}^{2 j + k + ℓ} \partial_{x}^{γ} G_{t} ∥}_{q} \\ = o (t^{- (n / 2) (1 - 1 / q) - N / 2}) & (1.8) \end{array}$ as $t \to \infty$ , where $μ_{0 γ}$ and $μ_{1 γ}$ are defined by (1.6) , $α_{j, k}$ , $β_{ℓ}$ are defined by (1.5) and $[\frac{N}{2}]$ is defined by (1.3) .

Since the estimate (1.7) in Theorem 1.1 requires only minimal regularity in the right-hand side, we can apply it to obtain sharp estimates of the solutions of the nonlinear damped wave equation such as $\partial_{t}^{2} u - Δ u + \partial_{t} u = f (u) .$ (1.9) For the nonlinear problem (1.9), the results will be given in the separated paper [31].

Remark 1.3.

For the Cauchy problem (1.2), the decay estimate and the asymptotic expansion of the solution $v (t)$ by the Gauss kernel are well known.

Proposition 1.4 (Cf. [9,16,17]).

Let $n ⩾ 1$ and $N \in Z_{+}$ . Assume that $g \in L_{N}^{1} \cap L^{\infty}$ . Then there exists a unique solution $v (t)$ of (1.2) in the class $C ([0, \infty); L_{N}^{1} \cap L^{\infty})$ satisfying the estimates ${{∥ v (t) ∥}_{q} ⩽ C (1 + t)}^{- (n / 2) (1 - 1 / q)}, t ⩾ 0,$ and ${∥ v (t) - \sum_{| γ | ⩽ N} μ_{γ} \partial_{x}^{γ} G_{t} ∥}_{q} = o (t^{- (n / 2) (1 - 1 / q) - N / 2})$ (1.10) as $t \to \infty$ for $1 ⩽ q ⩽ \infty$ , where $μ_{γ}$ are defined by $μ_{γ} : = \frac{1}{γ!} \int_{R^{n}} {(- y)}^{γ} g (y) d y$ (1.11) for $γ \in Z_{+}^{n}$ .

Remark 1.5.

Especially, the third-order expansion of the solution of (1.1) by the Gauss kernel is given by $\begin{array}{rcl} {∥ u (t) - \sum_{| γ | ⩽ 2} (μ_{0 γ} + μ_{1 γ}) \partial_{x}^{γ} G_{t} - t (μ_{00} + μ_{10}) Δ^{2} G_{t} - 2 (\frac{1}{2} μ_{00} + μ_{10}) (- Δ) G_{t} ∥}_{q} \\ = o (t^{- (n / 2) (1 - 1 / q) - 1}) \end{array}$ as $t \to \infty$ for $1 ⩽ q ⩽ \infty$ . On the other hand, the third-order expansion of the solution of (1.2) with the data $g = g_{0} + g_{1}$ ( $N = 2$ in the estimate (1.10)) is given by ${∥ v (t) - \sum_{| γ | ⩽ 2} (μ_{0 γ} + μ_{1 γ}) \partial_{x}^{γ} G_{t} ∥}_{q} = o (t^{- (n / 2) (1 - 1 / q) - 1})$ as $t \to \infty$ for $1 ⩽ q ⩽ \infty$ . Then we can observe the difference between the asymptotic behavior of the solution of (1.1) and that of (1.2).

Here we recall that the solution formula of (1.1) which is given by the following Fourier multiplier representation (see e.g. [13]) $u (t) = K_{0} (t) g_{0} + K_{1} (t) (\frac{1}{2} g_{0} + g_{1}),$ (1.12) where the evolution operators $K_{0} (t)$ and $K_{1} (t)$ are given by $\begin{array}{rclr} K_{0} (t) g : = F^{- 1} [e^{- t / 2} cos (t \sqrt{{| ξ |}^{2} - \frac{1}{4}}) \hat{g}], & (1.13) \\ K_{1} (t) g : = F^{- 1} [e^{- t / 2} \frac{sin (t \sqrt{{| ξ |}^{2} - 1 / 4})}{\sqrt{{| ξ |}^{2} - 1 / 4}} \hat{g}] . & (1.14) \end{array}$ Here we denote the Fourier and Fourier inverse transform by $F$ and $F^{- 1}$ . We also denote the Fourier transform of g by $\hat{g}$ .

The crucial part of the demonstrating the estimate (1.7) is the construction of the higher-order approximations of the operators $K_{0} (t)$ and $K_{1} (t)$ in the Fourier space. Namely, the approximations $\begin{matrix} \hat{K_{0} (t) g} = \frac{1}{2} \sum_{j = 0}^{[N / 2]} \sum_{k = 0}^{[N / 2] - j} α_{j, k} {(- t)}^{j} {| ξ |}^{2 (2 j + k)} e^{- t {| ξ |}^{2}} \hat{g} + error, \\ \hat{K_{1} (t) g} = \sum_{j = 0}^{[N / 2]} \sum_{k = 0}^{[N / 2] - j} \sum_{ℓ = 0}^{[N / 2] - j - k} α_{j, k} β_{ℓ} {(- t)}^{j} {| ξ |}^{2 (2 j + k + ℓ)} e^{- t {| ξ |}^{2}} \hat{g} + error \end{matrix}$ (1.15) for small $| ξ |$ and the Fourier multiplier theory enable us to obtain the higher-order expansion of the solutions for the linear problem (1.1) (see also the results [7,12] and [30] for the study of the asymptotic behavior of the solution to the linear problem (1.1) which has the intersection with the main topics in this paper).

The second important part is to establish the weighted $L^{1}$ estimate ${{∥ | x |^{m} u (t) ∥}_{1} ⩽ C (1 + t)}^{m / 2} ({∥ g_{0} ∥}_{W_{m}^{[n / 2], 1}} + {∥ g_{1} ∥}_{L_{m}^{1}}),$ (1.16) where u is the solution of (1.1) and $m ⩾ 0$ . The estimate (1.16) shows some consistency criterion. That is, if the data $(g_{0}, g_{1})$ is in $W_{m}^{[n / 2], 1} \times L_{m}^{1}$ then $u (t)$ is also in $L_{m}^{1}$ for all $t ⩾ 0$ .

Here we remark that Kawakami and Ueda [19] have obtained the estimate (1.16) by the different procedure. We will give another proof. We also note that in [19] they proved that the asymptotic profile of the solution to the nonlinear problem (1.9) coincides with that of the solution to the corresponding nonlinear parabolic problem $\partial_{t} u - Δ u = f (u),$ up to the second-order expansion, using the estimate (1.16) in combination with the well-known estimates $\begin{matrix} {∥ K_{0} (t) g_{0} - \frac{1}{2} e^{t Δ} g_{0} ∥}_{q} ⩽ C t^{- (n / 2) (1 / r - 1 / q) - 1} {∥ g_{0} ∥}_{r} + C e^{- δ t} {∥ g_{0} ∥}_{W^{[n / 2], q}}, t > 0, \\ {∥ K_{1} (t) g_{1} - e^{t Δ} g_{1} ∥}_{q} ⩽ C t^{- (n / 2) (1 / r - 1 / q) - 1} {∥ g_{1} ∥}_{r} + C e^{- δ t} {∥ g_{1} ∥}_{q}, t > 0 \end{matrix}$ (1.17) for $1 ⩽ r ⩽ q ⩽ \infty$ , $n = 1, 2, 3$ , which are shown by [13,23,27] and [26]. By contrast, we can regard our approximation (1.15) as an improvement of the estimates (1.17). As a consequence, Theorem 1.1 and Corollary 1.2 imply the difference between the solution of (1.1) and that of the corresponding parabolic problem (1.2).

We conclude this section by giving some notation used in this paper. Let $\hat{f}$ denote the Fourier transform of f defined by $\hat{f} (ξ) : = c_{n} \int_{R^{n}} e^{- i x \cdot ξ} f (x) d x$ with $c_{n} = {(2 π)}^{- n / 2}$ . Also, let $F^{- 1} [f]$ or $\overset{ˇ}{f}$ denote the inverse Fourier transform. For $k \in N$ and $1 ⩽ p ⩽ \infty$ , let $W^{k, p} (R^{n})$ be the Sobolev space: $W^{k, p} (R^{n}) : = {f : R^{n} \to R; {∥ f ∥}_{W^{1, p} (R^{n})} : = \sum_{| γ | ⩽ k} {∥ \partial_{x}^{γ} f ∥}_{p} < \infty},$ where $L^{p} (R^{n})$ is the Lebesgue space as usual and $H^{k} (R^{n}) : = W^{k, 2} (R^{n})$ . ${\dot{H}}^{k} (R^{n})$ is the homogeneous Sobolev space defined by ${\dot{H}}^{k} (R^{n}) : = {f : R^{n} \to R; {∥ f ∥}_{{\dot{H}}^{k} (R^{n})} : = \sum_{| γ | = k} {∥ \partial_{x}^{γ} f ∥}_{2} < \infty} .$ For the notation of the function spaces, the domain $R^{n}$ is often abbreviated. In the following C denotes a positive constant. It may change from line to line.

The paper is organized as follows. Section 2 presents some preliminaries. In Section 3, we show an appropriate decomposition of the propagators $K_{0} (t) g$ and $K_{1} (t) g$ in the Fourier space. Section 4 is devoted to the proof of Theorem 1.1, which is based on the Fourier multiplier theory. In Section 5, we show the weighted $L^{1}$ estimate of the solution of (1.1). We also prove Corollary 1.2 by applying Theorem 1.1 and the weighted $L^{1}$ estimate (1.16).

2. Preliminaries

In this section, we prepare several lemmas for the proofs of our results.

2.1. Estimates for the evolution operators

We begin with mentioning the $L^{q}$ – $L^{r}$ estimates for the evolution operators $K_{0} (t)$ and $K_{1} (t)$ .

Lemma 2.1 ([13,23,26,27]).

Let $n = 1, 2, 3$ , $1 ⩽ r ⩽ q ⩽ \infty$ and $0 < δ < \frac{1}{2}$ . Assume that $(g_{0}, g_{1}) \in L^{r} \cap W^{[n / 2], q} \times L^{r} \cap L^{q}$ . Then there exists a constant $C > 0$ such that $\begin{matrix} {∥ K_{0} (t) g_{0} ∥}_{q} ⩽ {C (1 + t)}^{- (n / 2) (1 / r - 1 / q)} {∥ g_{0} ∥}_{r} + C e^{- δ t} {∥ g_{0} ∥}_{W^{[n / 2], q}}, t ⩾ 0, \\ {∥ K_{1} (t) g_{1} ∥}_{q} ⩽ {C (1 + t)}^{- (n / 2) (1 / r - 1 / q)} {∥ g_{1} ∥}_{r} + C e^{- δ t} {∥ g_{1} ∥}_{q}, t ⩾ 0 . \end{matrix}$ (2.1)

The following estimates are well-known estimates of the heat semi-group (see e.g. Giga, Giga and Saal [9]). Lemma 2.2.

Let $n ⩾ 1$ , $1 ⩽ q ⩽ \infty$ , $N, j \in Z_{+}$ and $γ_{1}, γ_{2} \in Z_{+}^{n}$ . Assume that $g \in L^{r}$ . Then there exists a constant $C > 0$ such that ${∥ \partial_{t}^{j} \partial_{x}^{γ_{1}} e^{t Δ} g ∥}_{q} ⩽ C t^{- (n / 2) (1 - 1 / q) - | γ_{1} | / 2 - j} {∥ g ∥}_{r}, t > 0 .$ (2.2) Moreover, if $g \in L_{N}^{1}$ , it follows that ${∥ \partial_{t}^{j} \partial_{x}^{γ_{1}} (e^{t Δ} g - \sum_{| γ_{2} | ⩽ N} μ_{γ} \partial_{x}^{γ_{2}} G_{t}) ∥}_{q} = o (t^{- (n / 2) (1 - 1 / q) - N / 2 - j - | γ_{1} | / 2})$ (2.3) as $t \to \infty$ , where $μ_{γ}$ is defined by (1.11) .

2.2. Fourier multiplier

For $f \in L^{2} \cap L^{p}$ , $1 ⩽ p ⩽ \infty$ , let $m (ξ)$ be the Fourier multiplier defined by $F^{- 1} [m \hat{f}] (x) = c_{n} \int_{R^{n}} e^{- i x \cdot ξ} m (ξ) \hat{f} (ξ) d ξ .$ We define $M_{p}$ as the class of the Fourier multiplier for $1 ⩽ p ⩽ \infty$ : $M_{p} : = {m : R^{n} \to R; there exists a constant A_{p} > 0 such that {∥ F^{- 1} [m \hat{f}] ∥}_{p} ⩽ A_{p} {∥ f ∥}_{p}} .$ For $m \in M_{p}$ , we let $M_{p} (m) : = sup_{f \neq 0} \frac{{∥ F^{- 1} [m \hat{f}] ∥}_{p}}{{∥ f ∥}_{p}} .$

The following lemma describes the inclusion among the class of multipliers.

Lemma 2.3.
Let $\frac{1}{p} + \frac{1}{p^{'}} = 1$ with $1 ⩽ p ⩽ p^{'} ⩽ \infty$ , and assume that $m \in M_{p}$ . Then $m \in M_{q}$ for all $p ⩽ q ⩽ p^{'}$ and $M_{q} (m) ⩽ M_{p} (m) .$

We use the Carleson–Beurling inequality, which is applied to show the $L^{p}$ boundedness of Fourier multipliers.
Lemma 2.4 (Carleson–Beurling’s inequality).

If $m \in H^{s}$ with $s > \frac{n}{2}$ , then $m \in M_{r}$ for all $1 ⩽ r ⩽ \infty$ . Moreover, there exists a constant $C > 0$ such that $M_{\infty} (m) ⩽ C {∥ m ∥}_{2}^{1 - n / (2 s)} {∥ m ∥}_{{\dot{H}}^{s}}^{n / (2 s)} .$

For the proof of Lemmas 2.3 and 2.4, see [4].

2.3. Useful formula

The next lemma is useful to obtain decay estimates.

Lemma 2.5.
( i) Let $n ⩾ 1$ , $t ⩾ 0$ , $1 ⩽ q ⩽ \infty$ and $a > 0$ . There exists a constant $C > 0$ (independent of t) such that ${∥ e^{- t | ξ |^{2}} | ξ |^{a} ∥}_{L^{q} (| ξ | ⩽ 1)} ⩽ C {(1 + t)}^{- n / (2 q) - a / 2} .$ (2.4)

( ii) Let $n, ℓ = 1, 2, 3$ , $t > 0$ , $C_{0} > 0$ and $m ⩾ 0$ . There exists a constant $C > 0$ (independent of t) such that $\int_{| y | ⩽ 1} \frac{{| y |}^{m} e^{- C_{0} t {| y |}^{2}}}{{(1 - {| y |}^{2})}^{ℓ / 4}} d y ⩽ C t^{- n / 2 - m / 2} .$ (2.5)
Proof.
The estimate (2.4) is well known (see e.g. Takeda and Yoshikawa [32]). Here we only prove the estimate (2.5). Observing that the integrand is radial, we see that $\int_{| y | ⩽ 1} \frac{{| y |}^{m} e^{- C_{0} t {| y |}^{2}}}{{(1 - {| y |}^{2})}^{ℓ / 4}} d y ⩽ C \int_{0}^{1} \frac{τ^{m + n - 1} e^{- C_{0} t τ^{2}}}{{(1 - τ)}^{ℓ / 4}} d τ .$ Changing the integral variable $ζ = {(1 - τ)}^{1 / 4}$ and using $0 ⩽ ζ^{3 - ℓ} ⩽ 1$ for $1 ⩽ ℓ ⩽ 3$ , we see that $\begin{array}{rcl} \int_{| y | ⩽ 1} \frac{{| y |}^{m} e^{- C_{0} t {| y |}^{2}}}{{(1 - {| y |}^{2})}^{ℓ / 4}} d y & ⩽ & C \int_{0}^{1} \frac{τ^{m + n - 1} e^{- C_{0} t τ^{2}}}{{(1 - τ)}^{ℓ / 4}} d τ \\ = & C \int_{0}^{1} {(1 - ζ^{4})}^{m + n - 1} e^{- C_{0} t {(1 - ζ^{4})}^{2}} ζ^{3 - ℓ} d τ \\ ⩽ & C \int_{0}^{1} {(1 - ζ)}^{m + n - 1} e^{- C_{0} t {(1 - ζ)}^{2}} d τ \\ = & C \int_{0}^{1} τ^{m + n - 1} e^{- C_{0} t τ^{2}} d τ ⩽ C t^{- n / 2 - m / 2}, \end{array}$ which is the desired estimate and the proof is complete. □

3. Decomposition of the propagators in the Fourier space

In this section, we use the Fourier splitting method to decompose the evolution operators $K_{0} (t)$ and $K_{1} (t)$ into low, middle and high frequency parts. In what follows, we introduce the cut-off functions which will be used in the proofs to aligned to each frequency parts, respectively. Let $χ_{L}$ , $χ_{M}$ and $χ_{H} \in C^{\infty} (R^{n})$ be radial cut-off functions satisfying that $\begin{array}{rcl} χ_{L} (| ξ |) = \{\begin{matrix} 1, & | ξ | ⩽ \frac{1}{8}, \\ 0, & | ξ | ⩾ \frac{1}{4}, \end{matrix} χ_{H} (| ξ |) = \{\begin{matrix} 1, & | ξ | ⩾ 2, \\ 0, & | ξ | ⩽ 1, \end{matrix} \\ χ_{M} (| ξ |) = 1 - χ_{L} (| ξ |) - χ_{H} (| ξ |) . \end{array}$ Then we have the following decomposition formulas of $K_{0} (t) g$ and $K_{1} (t) g$ .

Lemma 3.1.
Let $n ⩾ 1$ and $N \in Z_{+}$ . Then, $K_{0} (t) g$ defined by (1.13) satisfies $\begin{array}{rclr} K_{0} (t) g - \frac{1}{2} \sum_{j = 0}^{[N / 2]} \sum_{k = 0}^{[N / 2] - j} α_{j, k} {(- t)}^{j} {(- Δ)}^{2 j + k} e^{t Δ} g \\ = F^{- 1} [A_{0} (t) \hat{g}] + F^{- 1} [\frac{1}{2} e^{- t / 2 - t \sqrt{1 / 4 - | ξ |^{2}}} χ_{L} (| ξ |) \hat{g}] \\ + F^{- 1} [e^{- t / 2} cos (t \sqrt{{| ξ |}^{2} - \frac{1}{4}}) (\sum_{J = M, H} χ_{J} (| ξ |)) \hat{g}] \\ - F^{- 1} [\sum_{j = 0}^{[N / 2]} \sum_{k = 0}^{[N / 2] - j} α_{j, k} {(- t)}^{j} (\sum_{J = M, H} χ_{J} (| ξ |)) {| ξ |}^{2 (2 j + k)} e^{- t {| ξ |}^{2}} \hat{g}], & (3.1) \end{array}$ where $A_{0} (t) : = \frac{1}{2} χ_{L} (| ξ |) e^{- t {| ξ |}^{2}} (e^{- t {| ξ |}^{4} ϕ_{1} ({| ξ |}^{2})} - \sum_{j = 0}^{[N / 2]} \sum_{k = 0}^{[N / 2] - j} α_{j, k} {(- t)}^{j} {| ξ |}^{2 (2 j + k)}) .$ (3.2)
Lemma 3.2.
Let $n ⩾ 1$ and $N \in Z_{+}$ . Then $K_{1} (t) g$ defined by (1.14) satisfies $\begin{array}{rclr} K_{1} (t) g - \sum_{j = 0}^{[N / 2]} \sum_{k = 0}^{[N / 2] - j} \sum_{ℓ = 0}^{[N / 2] - j - k} α_{j, k} β_{ℓ} {(- t)}^{j} {(- Δ)}^{2 j + k + ℓ} e^{t Δ} g \\ = F^{- 1} [A_{1} (t) \hat{g}] - F^{- 1} [e^{- t / 2 - t \sqrt{1 / 4 - {| ξ |}^{2}}} ψ ({| ξ |}^{2}) χ_{L} (| ξ |) \hat{g}] \\ + F^{- 1} [\frac{e^{- t / 2} sin (t \sqrt{{| ξ |}^{2} - 1 / 4})}{\sqrt{{| ξ |}^{2} - 1 / 4}} (\sum_{J = M, H} χ_{J} (| ξ |)) \hat{g}] \\ - F^{- 1} [\sum_{j = 0}^{[N / 2]} \sum_{k = 0}^{[N / 2] - j} \sum_{ℓ = 0}^{[N / 2] - j - k} α_{j, k} β_{ℓ} {(- t)}^{j} (\sum_{J = M, H} χ_{J} (| ξ |)) {| ξ |}^{2 (2 j + k + ℓ)} e^{- t {| ξ |}^{2}} \hat{g}], & (3.3) \end{array}$ where $\begin{array}{rclr} A_{1} (t) & : = & χ_{L} (| ξ |) e^{- t {| ξ |}^{2}} \\ \times (e^{- t {| ξ |}^{4} ϕ_{1} ({| ξ |}^{2})} ψ ({| ξ |}^{2}) - \sum_{j = 0}^{[N / 2]} \sum_{k = 0}^{[N / 2] - j} \sum_{ℓ = 0}^{[N / 2] - j - k} α_{j, k} β_{ℓ} {(- t)}^{j} {| ξ |}^{2 (2 j + k + ℓ)}) . & (3.4) \end{array}$

Here we only show the proof of Lemma 3.1. Lemma 3.2 is proved by the same argument as in the proof of Lemma 3.1. Proof of Lemma 3.1.
At first, the direct calculation yields that $\begin{array}{rcl} - \frac{{| ξ |}^{2}}{1 / 2 + \sqrt{1 / 4 - {| ξ |}^{2}}} + {| ξ |}^{2} & = & {| ξ |}^{2} {\frac{1}{1 / 2 + \sqrt{1 / 4 - {| ξ |}^{2}}} - 1} = \frac{- {| ξ |}^{4}}{{(1 / 2 + \sqrt{1 / 4 - {| ξ |}^{2}})}^{2}} \\ = & - ϕ_{1} ({| ξ |}^{2}) {| ξ |}^{4} . \end{array}$ Therefore using $cosh y = \frac{e^{y} + e^{- y}}{2}$ , we see that $\begin{array}{rclr} K_{0} (t) g & = & F^{- 1} [e^{- t / 2} cosh (t \sqrt{\frac{1}{4} - {| ξ |}^{2}}) χ_{L} (| ξ |) \hat{g}] \\ + F^{- 1} [e^{- t / 2} cos (t \sqrt{{| ξ |}^{2} - \frac{1}{4}}) (\sum_{j = M, H} χ_{j} (| ξ |)) \hat{g}] \\ = & F^{- 1} [\frac{1}{2} e^{- t / 2 + t \sqrt{1 / 4 - {| ξ |}^{2}}} χ_{L} (| ξ |) \hat{g}] + F^{- 1} [\frac{1}{2} e^{- t / 2 - t \sqrt{1 / 4 - {| ξ |}^{2}}} χ_{L} (| ξ |) \hat{g}] \\ + F^{- 1} [e^{- t / 2} cos (t \sqrt{{| ξ |}^{2} - \frac{1}{4}}) (\sum_{j = M, H} χ_{j} (| ξ |)) \hat{g}] . & (3.5) \end{array}$ On the other hand, we can rewrite the second term in the left-hand side of (3.1) as $\begin{array}{rclr} \frac{1}{2} \sum_{j = 0}^{[N / 2]} \sum_{k = 0}^{[N / 2] - j} α_{j, k} {(- t)}^{j} {(- Δ)}^{2 j + k} e^{t Δ} g \\ = \frac{1}{2} F^{- 1} [\sum_{j = 0}^{[N / 2]} \sum_{k = 0}^{[N / 2] - j} α_{j, k} {(- t)}^{j} (\sum_{J = L, M, H} χ_{J} (| ξ |)) {| ξ |}^{2 (2 j + k)} e^{- t {| ξ |}^{2}} \hat{g}] . & (3.6) \end{array}$ Thus, taking a difference between (3.5) and (3.6), we obtain (3.1) and Lemma 3.1 follows. □

4. Proof of main theorem

Our purpose in this section is to prove Theorem 1.1. According to the definitions of $K_{0} (t) g$ and $K_{1} (t) g$ (1.13) and (1.14), it is sufficient to show the following proposition.

Proposition 4.1.
Let $n = 1, 2, 3$ , $N \in Z_{+}$ , $1 ⩽ r ⩽ q ⩽ \infty$ and $0 < δ < \frac{1}{2}$ . Then there exists a constant $C > 0$ such that $\begin{array}{rclr} {∥ K_{0} (t) g - \frac{1}{2} \sum_{j = 0}^{[N / 2]} \sum_{k = 0}^{[N / 2] - j} α_{j, k} {(- t)}^{j} {(- Δ)}^{2 j + k} e^{t Δ} g ∥}_{q} \\ ⩽ \{\begin{matrix} C t^{- (n / 2) (1 / r - 1 / q) - ([N / 2] + 1)} {∥ g ∥}_{r} + C e^{- δ t} {∥ g ∥}_{W^{[n / 2], q}}, & t ⩾ 1, \\ C t^{- (n / 2) (1 / r - 1 / q) - [N / 2]} {∥ g ∥}_{r} + C e^{- δ t} {∥ g ∥}_{W^{[n / 2], q}}, & 0 < t ⩽ 1, \end{matrix} & (4.1) \\ {∥ K_{1} (t) g - \sum_{j = 0}^{[N / 2]} \sum_{k = 0}^{[N / 2] - j} \sum_{ℓ = 0}^{[N / 2] - j - k} α_{j, k} β_{ℓ} {(- t)}^{j} {(- Δ)}^{2 j + k + ℓ} e^{t Δ} g ∥}_{q} \\ ⩽ \{\begin{matrix} C t^{- (n / 2) (1 / r - 1 / q) - ([N / 2] + 1)} {∥ g ∥}_{r} + C e^{- δ t} {∥ g ∥}_{q}, & t ⩾ 1, \\ C t^{- (n / 2) (1 / r - 1 / q) - [N / 2]} {∥ g ∥}_{r} + C e^{- δ t} {∥ g ∥}_{q}, & 0 < t ⩽ 1 . \end{matrix} & (4.2) \end{array}$

Once we obtain Proposition 4.1, we immediately have Theorem 1.1. Proof of Theorem 1.1.
We apply Proposition 4.1 to the representation (1.12) and we obtain the estimate (1.7). The proof of Theorem 1.1 is complete. □
Proof of Proposition 4.1.
When $0 < t ⩽ 1$ , the linear estimates (2.1) and (2.2) show that $\begin{array}{rclr} {∥ K_{0} (t) g - \frac{1}{2} \sum_{j = 0}^{[N / 2]} \sum_{k = 0}^{[N / 2] - j} α_{j, k} {(- t)}^{j} {(- Δ)}^{2 j + k} e^{t Δ} g ∥}_{q} \\ ⩽ {∥ K_{0} (t) g ∥}_{q} + {∥ \frac{1}{2} \sum_{j = 0}^{[N / 2]} \sum_{k = 0}^{[N / 2] - j} α_{j, k} {(- t)}^{j} {(- Δ)}^{2 j + k} e^{t Δ} g ∥}_{q} \\ ⩽ C {(1 + t)}^{- (n / 2) (1 / r - 1 / q)} {∥ g ∥}_{r} + C e^{- δ t} {{∥ g ∥}_{W^{[n / 2], q}} + C t}^{- (n / 2) (1 / r - 1 / q) - [N / 2]} {∥ g ∥}_{r} \\ ⩽ C t^{- (n / 2) (1 / r - 1 / q) - [N / 2]} {∥ g ∥}_{r} + C e^{- δ t} {∥ g ∥}_{W^{[n / 2], q}}, & (4.3) \\ {∥ K_{1} (t) g - \sum_{j = 0}^{[N / 2]} \sum_{k = 0}^{[N / 2] - j} \sum_{ℓ = 0}^{[N / 2] - j - k} α_{j, k} β_{ℓ} {(- t)}^{j} {(- Δ)}^{2 j + k + ℓ} e^{t Δ} g ∥}_{q} \\ ⩽ {∥ K_{1} (t) g ∥}_{q} + {∥ \sum_{j = 0}^{[N / 2]} \sum_{k = 0}^{[N / 2] - j} \sum_{ℓ = 0}^{[N / 2] - j - k} α_{j, k} β_{ℓ} {(- t)}^{j} {(- Δ)}^{2 j + k + ℓ} e^{t Δ} g ∥}_{q} \\ ⩽ C {{(1 + t)}^{- (n / 2) (1 / r - 1 / q)} {∥ g ∥}_{r} + C e^{- δ t} {∥ g ∥}_{q} + C t}^{- (n / 2) (1 / r - 1 / q) - [N / 2]} {∥ g ∥}_{r} \\ ⩽ C t^{- (n / 2) (1 / r - 1 / q) - [N / 2]} {∥ g ∥}_{r} + C e^{- δ t} {∥ g ∥}_{q} . & (4.4) \end{array}$ Then we have (4.1) and (4.2) for $0 < t ⩽ 1$ .

In what follows, we assume $t ⩾ 1$ . Here we remark that the $L^{q}$ norms of the latter terms in (3.1) decay exponentially. Namely it is well known that $\begin{array}{rclr} {∥ F^{- 1} [e^{- t / 2 - t \sqrt{1 / 4 - {| ξ |}^{2}}} χ_{L} (| ξ |) \hat{g}] ∥}_{q} + {∥ F^{- 1} [cos (t \sqrt{{| ξ |}^{2} - \frac{1}{4}}) (\sum_{j = M, H} χ_{j} (ξ)) \hat{g}] ∥}_{q} \\ + {∥ \sum_{j = 0}^{[N / 2]} \sum_{k = 0}^{[N / 2] - j} α_{j, k} {(- t)}^{j} F^{- 1} [(\sum_{J = M, H} χ_{J}) {| ξ |}^{2 (2 j + k + ℓ)} e^{- t {| ξ |}^{2}} \hat{g}] ∥}_{q} ⩽ C e^{- δ t} {∥ g ∥}_{W^{[n / 2], q}} & (4.5) \end{array}$ for $1 ⩽ q ⩽ \infty$ , $t ⩾ 1$ (cf. [13,23,26,27]). In the same manner, we can see that in the right-hand side of (3.3), the $L^{q}$ norms of the latter three terms decay exponentially (cf. [13,23,26,27]). Namely, it is also well known that $\begin{array}{rclr} {∥ F^{- 1} [e^{- t / 2 - t \sqrt{1 / 4 - | ξ |^{2}}} ψ ({| ξ |}^{2}) χ_{L} (| ξ |) \hat{g}] ∥}_{q} \\ + {∥ F^{- 1} [\frac{e^{- t / 2} sin (t \sqrt{{| ξ |}^{2} - 1 / 4})}{\sqrt{{| ξ |}^{2} - 1 / 4}} (\sum_{J = M, H} χ_{J} (| ξ |)) \hat{g}] ∥}_{q} \\ + {∥ \sum_{j = 0}^{[N / 2]} \sum_{k = 0}^{[N / 2] - j} \sum_{ℓ = 0}^{[N / 2] - j - k} α_{j, k} β_{ℓ} {(- t)}^{j} F^{- 1} [(\sum_{J = M, H} χ_{J} (| ξ |)) {| ξ |}^{2 (2 j + k + ℓ)} e^{- t {| ξ |}^{2}} \hat{g}] ∥}_{q} \\ ⩽ C e^{- δ t} {∥ g ∥}_{q} & (4.6) \end{array}$ for $1 ⩽ q ⩽ \infty$ , $t ⩾ 1$ .

By this observation, the proof of Proposition 4.1 is reduced to the following lemma.
Lemma 4.2.
Let $n = 1, 2, 3$ , $N \in Z_{+}$ , $t ⩾ 1$ and $1 ⩽ r ⩽ q ⩽ \infty$ . Then there exists a constant $C > 0$ such that $\begin{array}{rclr} {∥ F^{- 1} [A_{0} (t, ξ) \hat{g}] ∥}_{q} ⩽ C t^{- (n / 2) (1 / r - 1 / q) - ([N / 2] + 1)} {∥ g ∥}_{r}, & (4.7) \\ {∥ F^{- 1} [A_{1} (t, ξ) \hat{g}] ∥}_{q} ⩽ C t^{- (n / 2) (1 / r - 1 / q) - ([N / 2] + 1)} {∥ g ∥}_{r}, & (4.8) \end{array}$ where $A_{0} (t, ξ)$ and $A_{1} (t, ξ)$ are defined by (3.2) and (3.4) respectively, $α_{j, k}$ and $β_{ℓ}$ are defined by (1.5) .
Proof.
For our purpose, we claim that $\begin{array}{rclr} {∥ F^{- 1} [A_{0} (t, ξ) \hat{g}] ∥}_{\infty} ⩽ C t^{- n / 2 - ([N / 2] + 1)} {∥ g ∥}_{1}, & (4.9) \\ {∥ F^{- 1} [A_{0} (t, ξ) \hat{g}] ∥}_{q} ⩽ C t^{- ([N / 2] + 1)} {∥ g ∥}_{q}, & (4.10) \\ {∥ F^{- 1} [A_{1} (t, ξ) \hat{g}] ∥}_{\infty} ⩽ C t^{- n / 2 - ([N / 2] + 1)} {∥ g ∥}_{1}, & (4.11) \\ {∥ F^{- 1} [A_{1} (t, ξ) \hat{g}] ∥}_{q} ⩽ C t^{- ([N / 2] + 1)} {∥ g ∥}_{q} & (4.12) \end{array}$ for $t ⩾ 1$ . Indeed, once we have the estimates (4.9)–(4.12), the estimates (4.9), (4.10) and the Riesz–Thorin interpolation theorem (cf. [3]) show the estimate (4.7). Similarly, the estimates (4.11), (4.12) and the Riesz–Thorin interpolation theorem directly yield the estimate (4.8).

At first, we show the estimate (4.9). By Taylor’s expansion, the integrand is divided into two parts: $F^{- 1} [A_{0} (t, ξ) \hat{g}] = A_{1} (t) + A_{2} (t) .$ Here we define $A_{J} (t)$ ( $J = 1, 2$ ) by $\begin{matrix} A_{1} (t) : = F^{- 1} [\frac{e^{- t {| ξ |}^{2}}}{2} (e^{- t {| ξ |}^{4} ϕ_{1} ({| ξ |}^{2})} - \sum_{j = 0}^{[N / 2]} \frac{{(- t {| ξ |}^{4} ϕ_{1} ({| ξ |}^{2}))}^{j}}{j!}) χ_{L} (| ξ |) \hat{g}], \\ A_{2} (t) : = F^{- 1} [\frac{e^{- t | ξ |^{2}}}{2} \sum_{j = 0}^{[N / 2]} \frac{{(- t {| ξ |}^{4})}^{j}}{j!} (ϕ_{j} ({| ξ |}^{2}) - \sum_{k = 0}^{[N / 2] - j} α_{j, ℓ} j! {| ξ |}^{2 ℓ}) χ_{L} (| ξ |) \hat{g}], \end{matrix}$ (4.13) where $ϕ_{j} ({| ξ |}^{2})$ is defined by (1.4) and $ϕ_{j} ({| ξ |}^{2})$ satisfies $ϕ_{j} ({| ξ |}^{2}) = ϕ_{1} {({| ξ |}^{2})}^{j}$ .

Thus we see that there exist some $θ_{1}, θ_{2} \in [0, 1]$ such that $\begin{array}{rclr} e^{- t {| ξ |}^{4} ϕ_{1} ({| ξ |}^{2})} - \sum_{j = 0}^{[N / 2]} \frac{{(- t {| ξ |}^{4} ϕ_{1} ({| ξ |}^{2}))}^{j}}{j!} = e^{- t θ_{1} {| ξ |}^{4} ϕ_{1} ({| ξ |}^{2})} \frac{{(- t {| ξ |}^{4} ϕ_{1} ({| ξ |}^{2}))}^{[N / 2] + 1}}{([N / 2] + 1)!}, & (4.14) \\ ϕ_{j} ({| ξ |}^{2}) - \sum_{k = 0}^{[N / 2] - j} α_{j, k} j! {| ξ |}^{2 k} = \frac{j!}{([N / 2] - j + 1)!} \frac{d^{[N / 2] - j + 1}}{d r^{[N / 2] - j + 1}} ϕ_{j} (r) |_{r = θ_{2} {| ξ |}^{2}} {| ξ |}^{2 ([N / 2] - j + 1)} & (4.15) \end{array}$ and then we can rewrite $A_{J} (t)$ ( $J = 1, 2$ ) as $\begin{array}{rcl} A_{1} (t) = \frac{{(- t)}^{[N / 2] + 1}}{2 ([N / 2] + 1)!} F^{- 1} [e^{- t {| ξ |}^{2} - t θ_{1} {| ξ |}^{4} ϕ_{1} ({| ξ |}^{2})} {| ξ |}^{4 ([N / 2] + 1)} ϕ_{[N / 2] + 1} ({| ξ |}^{2}) χ_{L} (| ξ |) \hat{g}], \\ A_{2} (t) = \sum_{j = 0}^{[N / 2]} \frac{{(- t)}^{j}}{2 ([N / 2] - j + 1)!} F^{- 1} [e^{- t {| ξ |}^{2}} \frac{d^{[N / 2] - j + 1}}{d r^{[N / 2] - j + 1}} ϕ_{j} (r) |_{r = θ_{2} {| ξ |}^{2}} {| ξ |}^{2 ([N / 2] + j + 1)} χ_{L} (| ξ |) \hat{g}] . \end{array}$ Then, for ${∥ A_{1} (t) ∥}_{\infty}$ , it follows from the Hausdorff–Young inequality and (2.4) that $\begin{array}{rclr} {∥ A_{1} (t) ∥}_{\infty} & ⩽ & C t^{[N / 2] + 1} {∥ e^{- t {| ξ |}^{2} - t θ_{1} {| ξ |}^{4} ϕ_{1} ({| ξ |}^{2})} {| ξ |}^{4 ([N / 2] + 1)} ϕ_{[N / 2] + 1} ({| ξ |}^{2}) χ_{L} (| ξ |) \hat{g} ∥}_{1} \\ ⩽ & C t^{[N / 2] + 1} {∥ e^{- t {| ξ |}^{2}} {| ξ |}^{4 ([N / 2] + 1)} χ_{L} (| ξ |) ∥}_{1} {∥ g ∥}_{1} \\ ⩽ & C t^{[N / 2] + 1} {(1 + t)}^{- n / 2 - 2 ([N / 2] + 1)} {∥ g ∥}_{1} ⩽ C {(1 + t)}^{- n / 2 - ([N / 2] + 1)} {∥ g ∥}_{1}, & (4.16) \end{array}$ where we have used the fact that there exists a constant $C > 0$ (independent of ξ) such that $| ϕ_{[N / 2] + 1} ({| ξ |}^{2}) | ⩽ C .$ (4.17) To show the estimate of ${∥ A_{2} (t) ∥}_{\infty}$ , we also remark that there exists a constant $C > 0$ (independent of ξ) such that $| \frac{d^{[N / 2] - j + 1}}{d r^{[N / 2] - j + 1}} ϕ_{j} (r) |_{r = θ_{2} {| ξ |}^{2}} | ⩽ C .$ (4.18) Applying the Hausdorff–Young inequality, (2.4) and (4.18), we see that $\begin{array}{rclr} {∥ A_{2} (t) ∥}_{\infty} & ⩽ & C \sum_{j = 0}^{[N / 2]} t^{j} {∥ e^{- t {| ξ |}^{2}} \frac{d^{[N / 2] - j + 1}}{d r^{[N / 2] - j + 1}} ψ_{j} (r) |_{r = θ_{2} {| ξ |}^{2}} {| ξ |}^{2 ([N / 2] + j + 1)} χ_{L} (| ξ |) \hat{g} ∥}_{1} \\ ⩽ & C \sum_{j = 0}^{[N / 2]} t^{j} {∥ e^{- t {| ξ |}^{2}} {| ξ |}^{2 ([N / 2] + j + 1)} χ_{L} (| ξ |) ∥}_{1} {∥ g ∥}_{1} \\ ⩽ & C \sum_{j = 0}^{[N / 2]} t^{j} {(1 + t)}^{- n / 2 - ([N / 2] + j + 1)} {∥ g ∥}_{1} ⩽ C {(1 + t)}^{- n / 2 - [N / 2] - 1} {∥ g ∥}_{1} . & (4.19) \end{array}$ Combining the estimates (4.16) and (4.19), we get the estimate (4.9).

Secondly, we prove the estimate (4.11). To do so, for $J = 1, 2, 3$ , we let $\begin{matrix} {\tilde{A}}_{1} (t) = \frac{{(- t)}^{[N / 2] + 1}}{([N / 2] + 1)!} F^{- 1} [e^{- t {| ξ |}^{2} - t θ_{1} {| ξ |}^{4} ϕ_{1} ({| ξ |}^{2})} {| ξ |}^{4 ([N / 2] + 1)} ϕ_{[N / 2] + 1} ({| ξ |}^{2}) ψ ({| ξ |}^{2}) χ_{L} (| ξ |) \hat{g}], \\ {\tilde{A}}_{2} (t) = \sum_{j = 0}^{[N / 2]} \frac{{(- t)}^{j}}{([N / 2] - j + 1)!} \\ \times F^{- 1} [e^{- t {| ξ |}^{2}} \frac{d^{[N / 2] - j + 1}}{d r^{[N / 2] - j + 1}} ϕ_{j} (r) |_{r = θ_{2} {| ξ |}^{2}} {| ξ |}^{2 ([N / 2] + j + 1)} ψ ({| ξ |}^{2}) χ_{L} (| ξ |) \hat{g}], \\ {\tilde{A}}_{3} (t) = F^{- 1} [e^{- t {| ξ |}^{2}} \sum_{j = 0}^{[N / 2]} \sum_{k = 0}^{[N / 2] - j} α_{j, k} {| ξ |}^{4 j + 2 k} {(- t)}^{j} (ψ ({| ξ |}^{2}) - \sum_{ℓ = 0}^{[N / 2] - j - k} β_{ℓ} {| ξ |}^{2 ℓ}) χ_{L} (| ξ |) \hat{g}] . \end{matrix}$ (4.20) Then the integrand $F^{- 1} [A_{1} (t, ξ) \hat{g}]$ ( $A_{1} (t, ξ)$ is defined by (3.4)) is divided into three parts: $F^{- 1} [A_{1} (t, ξ) \hat{g}] = \sum_{J = 1}^{3} {\tilde{A}}_{J} (t),$ where we use (4.14) and (4.15). By the estimate $| ψ ({| ξ |}^{2}) | ⩽ C$ and the similar fashion of the derivation of the estimates (4.16) and (4.19), we reached at the estimate ${∥ {\tilde{A}}_{1} (t) ∥}_{\infty} + {∥ {\tilde{A}}_{2} (t) ∥}_{\infty} ⩽ C {(1 + t)}^{- n / 2 - ([N / 2] + 1)} {∥ g ∥}_{1} .$ (4.21) The third term ${∥ {\tilde{A}}_{3} (t) ∥}_{\infty}$ is also estimated by the similar way. Indeed, we again use the Taylor expansion of ψ up to $([\frac{N}{2}] - j - k)$ th order and $\begin{array}{rcl} ψ ({| ξ |}^{2}) - \sum_{ℓ = 0}^{[N / 2] - j - k} β_{ℓ} {| ξ |}^{2 ℓ} \\ = \frac{1}{([N / 2] - j - k + 1)!} \frac{d^{[N / 2] - j - k + 1}}{d r^{[N / 2] - j - k + 1}} ψ (r) |_{r = θ_{3} {| ξ |}^{2}} {| ξ |}^{2 ([N / 2] - j - k + 1)} \end{array}$ for some $θ_{3} \in [0, 1]$ . Then we apply the Hausdorff–Young inequality and the estimate (2.4) to have $\begin{array}{rclr} {∥ {\tilde{A}}_{3} (t) ∥}_{\infty} & ⩽ & C \sum_{j = 0}^{[N / 2]} \sum_{k = 0}^{[N / 2] - j} t^{j} {∥ e^{- t {| ξ |}^{2}} \frac{d^{[N / 2] - j - k + 1}}{d r^{[N / 2] - j - k + 1}} ψ (r) |_{r = θ_{3} {| ξ |}^{2}} {| ξ |}^{2 ([N / 2] + j + 1)} χ_{L} (| ξ |) \hat{g} ∥}_{1} \\ ⩽ & C \sum_{j = 0}^{[N / 2]} t^{j} {∥ e^{- t {| ξ |}^{2}} {| ξ |}^{2 ([N / 2] + j + 1)} χ_{L} (| ξ |) ∥}_{1} {∥ g ∥}_{1} \\ ⩽ & C \sum_{j = 0}^{[N / 2]} t^{j} {(1 + t)}^{- n / 2 - [N / 2] - 1 - j} {∥ g ∥}_{1} ⩽ C {(1 + t)}^{- n / 2 - [N / 2] - 1} {∥ g ∥}_{1}, & (4.22) \end{array}$ where we have used the estimate $| \frac{d^{[N / 2] - j - ℓ + 1}}{d r^{[N / 2] - j - ℓ + 1}} ψ (r) |_{r = θ_{3} {| ξ |}^{2}} | ⩽ C$ for $C > 0$ , which is independent of ξ. Combining the estimates (4.21) and (4.22), we obtain the estimate (4.11).

Next we show the estimates (4.10) and (4.12) for $1 ⩽ q ⩽ \infty$ . In what follows, we denote Fourier multipliers in $A_{J} (t)$ ( $J = 1, 2$ ) (defined by (4.13)), ${\tilde{A}}_{J} (t)$ ( $J = 1, 2, 3$ ) (defined by (4.20)) by $\begin{array}{rclr} m_{0} (t, ξ) = e^{- t θ_{1} {| ξ |}^{4} ϕ_{1} ({| ξ |}^{2})} {| ξ |}^{4 ([N / 2] + 1)} ϕ_{[N / 2] + 1} ({| ξ |}^{2}) χ_{L} (| ξ |), \\ m_{0 j} (t, ξ) = \frac{d^{[N / 2] - j + 1}}{d r^{[N / 2] - j + 1}} ϕ_{j} (r) |_{r = θ_{2} {| ξ |}^{2}} {| ξ |}^{2 ([N / 2] + j + 1)} χ_{L} (| ξ |), \\ m_{1} (t, ξ) = ψ (| ξ |^{2}) m_{0} (t, ξ), & (4.23) \\ m_{1 j} (t, ξ) = ψ ({| ξ |}^{2}) m_{0 j} (t, ξ), \\ m_{1 j k} (t, ξ) = \frac{d^{[N / 2] - j - k + 1}}{d r^{[N / 2] - j - k + 1}} ψ (r) |_{r = θ_{3} {| ξ |}^{2}} {| ξ |}^{2 ([N / 2] + j + 1)} χ_{L} (| ξ |) . \end{array}$ Namely we rewrite $A_{J} (t)$ ( $J = 1, 2$ ) and ${\tilde{A}}_{J} (t)$ ( $J = 1, 2, 3$ ) as $\begin{array}{rclr} A_{1} (t) = \frac{{(- t)}^{[N / 2] + 1}}{2 ([N / 2] + 1)!} F^{- 1} [e^{- t {| ξ |}^{2}} m_{0} (t, ξ) \hat{g}], \\ A_{2} (t) = \sum_{j = 0}^{[N / 2]} \frac{{(- t)}^{j}}{2 ([N / 2] - j + 1)!} F^{- 1} [e^{- t {| ξ |}^{2}} m_{0 j} (t, ξ) \hat{g}], \\ {\tilde{A}}_{1} (t) = \frac{{(- t)}^{[N / 2] + 1}}{([N / 2] + 1)!} F^{- 1} [e^{- t {| ξ |}^{2}} m_{1} (t, ξ) \hat{g}], & (4.24) \\ {\tilde{A}}_{2} (t) = \sum_{j = 0}^{[N / 2]} \frac{{(- t)}^{j}}{([N / 2] - j + 1)!} F^{- 1} [e^{- t {| ξ |}^{2}} m_{1 j} (t, ξ) \hat{g}], \\ {\tilde{A}}_{3} (t) = \sum_{j = 0}^{[N / 2]} \sum_{k = 0}^{[N / 2] - j} \frac{{(- t)}^{j} α_{j, k}}{([N / 2] - j - k + 1)!} F^{- 1} [e^{- t {| ξ |}^{2}} m_{1 j k} (t, ξ) \hat{g}] . \end{array}$ Our strategy for the proof of (4.9) and (4.12) is the use of the Carleson–Beurling inequality (Lemma 2.4). Then we need to calculate the $L^{2}$ norm, ${\dot{H}}^{1}$ norm (for $n = 1$ ) and ${\dot{H}}^{2}$ norm (for $n = 2, 3$ ) of multipliers. That is, we claim that $\begin{array}{rclr} {∥ e^{- t | ξ |^{2}} m_{0} (t, ξ) ∥}_{{\dot{H}}^{ℓ}} ⩽ {C t}^{- n / 4 - 2 ([N / 2] + 1) + ℓ / 2}, & (4.25) \\ {∥ e^{- t {| ξ |}^{2}} m_{0 j} (t, ξ) ∥}_{{\dot{H}}^{ℓ}} ⩽ {C t}^{- n / 4 - ([N / 2] + 1 + j) + ℓ / 2}, & (4.26) \\ {∥ e^{- t {| ξ |}^{2}} m_{1} (t, ξ) ∥}_{{\dot{H}}^{ℓ}} ⩽ {C t}^{- n / 4 - 2 ([N / 2] + 1) + ℓ / 2}, & (4.27) \\ {∥ e^{- t {| ξ |}^{2}} m_{1 j} (t, ξ) ∥}_{{\dot{H}}^{ℓ}} ⩽ {C t}^{- n / 4 - ([N / 2] + 1 + j) + ℓ / 2}, & (4.28) \\ {∥ e^{- t {| ξ |}^{2}} m_{1 j k} (t, ξ) ∥}_{{\dot{H}}^{ℓ}} ⩽ {C t}^{- n / 4 - ([N / 2] + 1 + j) + ℓ / 2} & (4.29) \end{array}$ for $ℓ = 0, 1, 2$ , where we identify ${\dot{H}}^{0} (R^{n}) = L^{2} (R^{n})$ . We note that once we have the claim for $m_{0}$ and $m_{0 j}$ (4.25) and (4.26), the estimates for $m_{1}$ , $m_{1 j}$ and $m_{1 j k}$ (4.27), (4.28) and (4.29) immediately follow since $ψ (| ξ |^{2})$ satisfies that there exists a constant $C > 0$ (independent of ξ) such that $| ψ ({| ξ |}^{2}) | ⩽ C, | \partial_{ξ_{J}} \partial_{ξ_{K}} ψ ({| ξ |}^{2}) | ⩽ C$ (4.30) on the support of $χ_{L}$ . Thus we concentrate on the proof of estimates (4.25) and (4.26).

For the proof of (4.25), when $ℓ = 0$ , the estimate follows by the same methods as in the proofs of (4.16) and (4.19) since we have already shown the $L^{1}$ estimates of the multipliers $e^{- t {| ξ |}^{2}} m_{0}$ and $e^{- t {| ξ |}^{2}} m_{0 j}$ in the procedure of (4.16) and (4.19). For $ℓ = 2$ , we recall that ${∥ g ∥}_{{\dot{H}}^{2} (R^{n})} ⩽ C {∥ (- Δ) g ∥}_{2}$ for $g \in {\dot{H}}^{2}$ and $- Δ_{x} g = \partial_{| x |}^{2} g + \frac{n - 1}{| x |} \partial_{| x |} g$ when $g (x) = g (| x |)$ . The multipliers $e^{- t {| ξ |}^{2}} m_{0} (t, ξ)$ in $A_{1} (t)$ and $e^{- t {| ξ |}^{2}} m_{0 j} (t, ξ)$ in $A_{2} (t)$ are radial and then it suffices to show the estimates $\begin{array}{rclr} {∥ (\partial_{| ξ |}^{2} + \frac{n - 1}{| ξ |} \partial_{| ξ |}) (e^{- t {| ξ |}^{2}} m_{0} (t, ξ)) ∥}_{2} ⩽ {C t}^{- n / 4 - 2 ([N / 2] + 1) + 1}, & (4.31) \\ {∥ (\partial_{| ξ |}^{2} + \frac{n - 1}{| ξ |} \partial_{| ξ |}) (e^{- t {| ξ |}^{2}} m_{0 j} (t, ξ)) ∥}_{2} ⩽ {C t}^{- n / 4 - ([N / 2] + 1 + j) + 1} . & (4.32) \end{array}$ We denote $| ξ |$ briefly by τ. Noting that $\partial_{τ} e^{- t τ^{2}} = - 2 t τ e^{- t τ^{2}}$ and $\partial_{τ}^{2} e^{- t τ^{2}} = (4 t^{2} τ^{2} - 2 t) e^{- t τ^{2}}$ , we have $\begin{array}{rclr} (\partial_{τ}^{2} + \frac{n - 1}{τ} \partial_{τ}) (e^{- t τ^{2}} m_{0} (t, τ)) \\ = (\partial_{τ}^{2} e^{- t τ^{2}}) m_{0} (t, τ) + 2 (\partial_{τ} e^{- t τ^{2}}) \partial_{τ} m_{0} (t, τ) + e^{- t τ^{2}} \partial_{τ}^{2} m_{0} (t, τ) \\ + \frac{n - 1}{τ} ((\partial_{τ} e^{- t τ^{2}}) m_{0} (t, τ) + e^{- t τ^{2}} (\partial_{τ} m_{0} (t, τ))) \\ = e^{- t τ^{2}} {(4 t^{2} τ^{2} - 2 n t) m_{0} (t, τ) + (- 4 t τ + \frac{n - 1}{τ}) \partial_{τ} m_{0} (t, τ) + \partial_{τ}^{2} m_{0} (t, τ)} . & (4.33) \end{array}$ To obtain (4.31), we prepare point-wise estimates for $m_{0} (t, τ)$ , $\partial_{τ} m_{0} (t, τ)$ and $\partial_{τ}^{2} m_{0} (t, τ)$ . By the definition of $m_{0} (t, τ)$ (4.23) and the estimate (4.17), we see $| m_{0} (t, τ) | ⩽ C {| τ |}^{4 ([N / 2] + 1)} χ_{L} (τ) .$ (4.34) For $\partial_{τ} m_{0} (t, τ)$ , observing that $\begin{array}{rclr} \partial_{τ} m_{0} (t, τ) & = & \partial_{τ} (e^{- t θ_{1} τ^{4} ϕ_{1} (τ^{2})} τ^{4 ([N / 2] + 1)} ϕ_{[N / 2] + 1} (τ^{2}) χ_{L} (τ)) \\ = & \partial_{τ} (e^{- t θ_{1} τ^{4} ϕ_{1} (τ^{2})} τ^{4 ([N / 2] + 1)} ϕ_{[N / 2] + 1} (τ^{2})) χ_{L} (τ) \\ + e^{- t θ_{1} τ^{4} ϕ_{1} (τ^{2})} τ^{4 ([N / 2] + 1)} ϕ_{[N / 2] + 1} (τ^{2}) χ_{L}^{'} (τ) \\ = & e^{- t θ_{1} τ^{4} ϕ_{1} (τ^{2})} \partial_{τ} (- t θ_{1} τ^{4} ϕ_{1} (τ^{2})) τ^{4 ([N / 2] + 1)} ϕ_{[N / 2] + 1} (τ^{2}) χ_{L} (τ) \\ + e^{- t θ_{1} τ^{4} ϕ_{1} (τ^{2})} \partial_{τ} (τ^{4 ([N / 2] + 1)} ϕ_{[N / 2] + 1} (τ^{2})) χ_{L} (τ) \\ + e^{- t θ_{1} τ^{4} ϕ_{1} (τ^{2})} τ^{4 ([N / 2] + 1)} ϕ_{[N / 2] + 1} (τ^{2}) χ_{L}^{'} (τ), & (4.35) \end{array}$ we arrive at the estimate $| \partial_{τ} m_{0} (t, τ) | ⩽ C t τ^{4 [N / 2] + 7} χ_{L} (τ) + C τ^{4 [N / 2] + 3} χ_{L} (τ) + C χ_{L}^{'} (τ),$ (4.36) where we have used the estimates $| \partial_{τ} (τ^{4} ϕ_{1} (τ^{2})) | ⩽ C τ^{3}, | \partial_{τ} (τ^{4 ([N / 2] + 1)} {ϕ_{[N / 2] + 1} (τ^{2})) | ⩽ C τ}^{4 [N / 2] + 3}$ (4.37) on the support of $χ_{L}$ to estimate the first term and the second term in the right-hand side of (4.35).

To obtain the point-wise estimate for $\partial_{τ}^{2} m_{0} (t, τ)$ , we see that the direct calculation yields that $\begin{array}{rclr} \partial_{τ}^{2} m_{0} (t, τ) & = & \partial_{τ}^{2} (e^{- t θ_{1} τ^{4} ϕ_{1} (τ^{2})} τ^{4 ([N / 2] + 1)} ϕ_{[N / 2] + 1} (τ^{2})) χ_{L} (τ) \\ + 2 \partial_{τ} (e^{- t θ_{1} τ^{4} ϕ_{1} (τ^{2})} τ^{4 ([N / 2] + 1)} ϕ_{[N / 2] + 1} (τ^{2})) χ_{L}^{'} (τ) \\ + \partial_{τ} (e^{- t θ_{1} τ^{4} ϕ_{1} (τ^{2})} τ^{4 ([N / 2] + 1)} ϕ_{[N / 2] + 1} (τ^{2}) χ_{L}^{'} (τ)) \\ = & \partial_{τ}^{2} (e^{- t θ_{1} τ^{4} ϕ_{1} (τ^{2})}) τ^{4 ([N / 2] + 1)} ϕ_{[N / 2] + 1} (τ^{2}) χ_{L} (τ) \\ + 2 \partial_{τ} (e^{- t θ_{1} τ^{4} ϕ_{1} (τ^{2})}) \partial_{τ} (τ^{4 ([N / 2] + 1)} ϕ_{[N / 2] + 1} (τ^{2})) χ_{L} (τ) \\ + e^{- t θ_{1} τ^{4} ϕ_{1} (τ^{2})} \partial_{τ}^{2} (τ^{4 ([N / 2] + 1)} ϕ_{[N / 2] + 1} (τ^{2})) χ_{L} (τ) \\ + 2 \partial_{τ} (e^{- t θ_{1} τ^{4} ϕ_{1} (τ^{2})} τ^{4 ([N / 2] + 1)} ϕ_{[N / 2] + 1} (τ^{2})) χ_{L}^{'} (τ) \\ + \partial_{τ} (e^{- t θ_{1} τ^{4} ϕ_{1} (τ^{2})} τ^{4 ([N / 2] + 1)} ϕ_{[N / 2] + 1} (τ^{2}) χ_{L}^{'} (τ)) . & (4.38) \end{array}$ For the first term in the right-hand side of (4.38), we have $\begin{array}{rclr} | \partial_{τ}^{2} (e^{- t θ_{1} τ^{4} ϕ_{1} (τ^{2})}) | & ⩽ & (| t^{2} θ_{1}^{2} {(\partial_{τ} (τ^{4} ϕ_{1} (τ^{2})))}^{2} | + | t θ_{1} \partial_{τ}^{2} (τ^{4} ϕ_{1} (τ^{2})) |) e^{- t θ_{1} τ^{4} ϕ_{1} (τ^{2})} \\ ⩽ & C t^{2} τ^{6} + C t τ^{2} & (4.39) \end{array}$ on the support of $χ_{L}$ , since $\begin{array}{rcl} \partial_{τ}^{2} (e^{- t θ_{1} τ^{4} ϕ_{1} (τ^{2})}) & = & \partial_{τ} (e^{- t θ_{1} τ^{4} ϕ_{1} (τ^{2})}) \partial_{τ} (- t θ_{1} τ^{4} ϕ_{1} (τ^{2})) \\ = & e^{- t θ_{1} τ^{4} ϕ_{1} (τ^{2})} {{(\partial_{τ} (- t θ_{1} τ^{4} ϕ_{1} (τ^{2})))}^{2} + \partial_{τ}^{2} (- t θ_{1} τ^{4} ϕ_{1} (τ^{2}))} . \end{array}$ Thus by the estimate (4.39) and the similar method as in (4.37), we have $\begin{array}{rclr} | \partial_{τ}^{2} m_{0} (t, τ) | & ⩽ & C (t^{2} τ^{6} + t τ^{2}) τ^{4 ([N / 2] + 1)} χ_{L} (τ) + C t τ^{4 [N / 2] + 7} χ_{L} (τ) \\ + C τ^{4 [N / 2] + 2} χ_{L} (τ) + C | χ_{L}^{'} (τ) | + C | χ_{L}^{″} (τ) | & (4.40) \end{array}$ on the support of $χ_{L}$ . Combining (4.33) and the estimates (4.34), (4.36) and (4.40), we obtain $\begin{array}{rcl} | (\partial_{τ}^{2} + \frac{n - 1}{τ} \partial_{τ}) (e^{- t τ^{2}} m_{0} (t, τ)) | \\ = e^{- t τ^{2}} | (4 t^{2} τ^{2} - 2 n t) m_{0} (t, τ) + (- 4 t τ + \frac{n - 1}{τ}) \partial_{τ} m_{0} (t, τ) + \partial_{τ}^{2} m_{0} (t, τ) | \\ ⩽ C e^{- t τ^{2}} (t^{2} τ^{2} + t) τ^{4 ([N / 2] + 1)} χ_{L} (τ) \\ + C e^{- t τ^{2}} (t τ + \frac{1}{τ}) (t τ^{4 [N / 2] + 7} χ_{L} (τ) + τ^{4 [N / 2] + 3} χ_{L} (τ) + | χ_{L}^{'} (τ) |) \\ + C e^{- t τ^{2}} {(t^{2} τ^{6} + t τ^{2}) τ^{4 ([N / 2] + 1)} χ_{L} (τ) \\ + t τ^{4 [N / 2] + 7} χ_{L} (τ) + τ^{4 [N / 2] + 2} χ_{L} (τ) + | χ_{L}^{'} (τ) | + | χ_{L}^{″} (τ) |} \\ ⩽ C e^{- t τ^{2}} {(t^{2} τ^{2} + t) τ^{4 ([N / 2] + 1)} χ_{L} (τ) + τ^{4 [N / 2] + 2} χ_{L} (τ) + | χ_{L}^{'} (τ) | + | χ_{L}^{″} (τ) |} \end{array}$ on the support of $χ_{L}$ . Therefore, by the estimate (2.4), it is easy to verify that $\begin{array}{rcl} {∥ (\partial_{| ξ |}^{2} + \frac{n - 1}{| ξ |} \partial_{| ξ |}) (e^{- t {| ξ |}^{2}} m_{0} (t, ξ)) ∥}_{2} \\ ⩽ C {∥ e^{- t {| ξ |}^{2}} {(t^{2} {| ξ |}^{2} + t) {| ξ |}^{4 ([N / 2] + 1)} χ_{L} (| ξ |) + {| ξ |}^{4 [N / 2] + 2} χ_{L} (| ξ |)} ∥}_{2} + C e^{- δ t} \\ ⩽ C t^{2} {∥ e^{- t | ξ |^{2}} {| ξ |}^{4 [N / 2] + 6} χ_{L} (| ξ |) ∥}_{2} + C t {∥ e^{- t {| ξ |}^{2}} {| ξ |}^{4 [N / 2] + 4} χ_{L} (| ξ |) ∥}_{2} \\ + {∥ e^{- t {| ξ |}^{2}} {| ξ |}^{4 [N / 2] + 2} χ_{L} (| ξ |) ∥}_{2} + C e^{- δ t} \\ ⩽ C t^{- n / 4 - 2 [N / 2] - 1}, \end{array}$ where we remark that the supports of $χ_{L}^{'}$ and $χ_{L}^{″}$ do not include the neighborhood of the origin $ξ = 0$ . The estimate (4.25) for $ℓ = 1$ is easily shown by the interpolation inequality ${∥ g ∥}_{{\dot{H}}^{1}} ⩽ C {∥ g ∥}_{2}^{1 / 2} {∥ g ∥}_{{\dot{H}}^{2}}^{1 / 2} .$ (4.41) Then we have proved the estimate (4.25) for $ℓ = 0, 1, 2$ .

For the proof of (4.32), by the similar method as in (4.31), we also have $\begin{array}{rcl} (\partial_{τ}^{2} + \frac{n - 1}{τ} \partial_{τ}) (e^{- t τ^{2}} m_{0 j} (t, τ)) \\ = e^{- t τ^{2}} {(4 t^{2} τ^{2} - 2 n t) m_{0 j} (t, τ) + (\frac{n - 1}{τ} - 4 t τ) \partial_{τ} m_{0 j} (t, τ) + \partial_{τ}^{2} m_{0 j} (t, τ)}, \end{array}$ where $m_{0 j}$ is defined by (4.23). In addition, the direct calculation yields that $\begin{array}{rcl} \partial_{τ} m_{0 j} (t, τ) = \partial_{τ} (\frac{d^{[N / 2] - j + 1}}{d r^{[N / 2] - j + 1}} ϕ_{j} (r) |_{r = θ_{2} τ^{2}} τ^{2 ([N / 2] + j + 1)} χ_{L} (τ)) \\ = 2 θ_{2} \frac{d^{[N / 2] - j + 2}}{d r^{[N / 2] - j + 2}} ϕ_{j} (r) |_{r = θ_{2} τ^{2}} τ^{2 ([N / 2] + j + 1) + 1} χ_{L} (τ) \\ + 2 ([\frac{N}{2}] + j + 1) \frac{d^{[N / 2] - j + 1}}{d r^{[N / 2] - j + 1}} ϕ_{j} (r) |_{r = θ_{2} τ^{2}} τ^{2 ([N / 2] + j) + 1} χ_{L} (τ) \\ + \frac{d^{[N / 2] - j + 1}}{d r^{[N / 2] - j + 1}} ϕ_{j} (r) |_{r = θ_{2} τ^{2}} τ^{2 ([N / 2] + j + 1)} χ_{L}^{'} (τ), \\ \partial_{τ}^{2} m_{0 j} (t, τ) = \partial_{τ}^{2} (\frac{d^{[N / 2] - j + 1}}{d r^{[N / 2] - j + 1}} ϕ_{j} (r) |_{r = θ_{2} τ^{2}} τ^{2 ([N / 2] + j + 1)} χ_{L} (τ)) \\ = 4 θ_{2}^{2} \frac{d^{[N / 2] - j + 3}}{d r^{[N / 2] - j + 3}} ϕ_{j} (r) |_{r = θ_{2} τ^{2}} τ^{2 ([N / 2] + j + 2)} χ_{L} (τ) \\ + 2 θ_{2} (2 ([\frac{N}{2}] + j + 1) + 1) \frac{d^{[N / 2] - j + 2}}{d r^{[N / 2] - j + 2}} ϕ_{j} (r) |_{r = θ_{2} τ^{2}} τ^{2 ([N / 2] + j + 1)} χ_{L} (τ) \\ + 4 θ_{2} ([\frac{N}{2}] + j + 1) \frac{d^{[N / 2] - j + 2}}{d r^{[N / 2] - j + 2}} ϕ_{j} (r) |_{r = θ_{2} τ^{2}} τ^{2 ([N / 2] + j + 1)} χ_{L} (τ) \\ + 2 ([\frac{N}{2}] + j + 1) \\ \times (2 ([\frac{N}{2}] + j) + 1) \frac{d^{[N / 2] - j + 1}}{d r^{[N / 2] - j + 1}} ϕ_{j} (r) |_{r = θ_{2} τ^{2}} τ^{2 ([N / 2] + j)} χ_{L} (τ) \\ + {2 θ_{2} \frac{d^{[N / 2] - j + 2}}{d r^{[N / 2] - j + 2}} ϕ_{j} (r) |_{r = θ_{2} τ^{2}} τ^{2 ([N / 2] + j + 1) + 1} \\ + 2 ([\frac{N}{2}] + j + 1) \frac{d^{[N / 2] - j + 1}}{d r^{[N / 2] - j + 1}} ϕ_{j} (r) |_{r = θ_{2} τ^{2}} τ^{2 ([N / 2] + j) + 1}} χ_{L}^{'} (τ) \\ + \partial_{τ} (\frac{d^{[N / 2] - j + 2}}{d r^{[N / 2] - j + 2}} ϕ_{j} (r) |_{r = θ_{2} τ^{2}} τ^{2 ([N / 2] + j + 1)} χ_{L}^{'} (τ)) . \end{array}$

Then as in the proof of the estimates (4.34), (4.36) and (4.40), we can assert that $\begin{array}{rcl} | m_{0 j} (t, τ) | ⩽ C τ^{2 ([N / 2] + j + 1)} χ_{L} (τ), \\ | \partial_{τ} m_{0 j} (t, τ) | ⩽ C τ^{2 ([N / 2] + j) + 1} χ_{L} (τ) + C χ_{L}^{'} (τ), \\ | \partial_{τ}^{2} m_{0 j} (t, τ) | ⩽ C {τ^{2 ([N / 2] + j + 2)} + C τ}^{2 ([N / 2] + j + 1)} χ_{L} (τ) \\ {+ C τ}^{2 ([N / 2] + j)} χ_{L} (τ) + C χ_{L}^{'} (τ) + C χ_{L}^{″} (τ) \\ {⩽ C τ}^{2 ([N / 2] + j)} χ_{L} (τ) + C χ_{L}^{'} (τ) + C χ_{L}^{″} (τ) . \end{array}$

Combining these inequalities gives $\begin{array}{rcl} | (\partial_{τ}^{2} + \frac{n - 1}{τ} \partial_{τ}) (e^{- t τ^{2}} m_{0 j} (t, τ)) | \\ ⩽ C e^{- t τ^{2}} (t^{2} τ^{2} + t) τ^{2 ([N / 2] + j + 1)} χ_{L} (τ) + C e^{- t τ^{2}} (t τ + \frac{1}{τ}) (τ^{2 ([N / 2] + j) + 1} χ_{L} (τ) + | χ_{L}^{'} (τ) |) \\ + C e^{- t τ^{2}} (τ^{2 ([N / 2] + j)} + | χ_{L}^{'} (τ) | + | χ_{L}^{″} (τ) |) \\ ⩽ C e^{- t τ^{2}} (t^{2} τ^{2} + t) τ^{2 ([N / 2] + j + 1)} χ_{L} (τ) + C e^{- t τ^{2}} τ^{2 ([N / 2] + j)} χ_{L} (τ) \\ + C e^{- t δ τ^{2}} (| χ_{L}^{'} (τ) | + | χ_{L}^{″} (τ) |) . \end{array}$ Therefore we see that $\begin{array}{rcl} {∥ (\partial_{| ξ |}^{2} + \frac{n - 1}{| ξ |} \partial_{| ξ |}) (e^{- t {| ξ |}^{2}} m_{0 j} (t, ξ)) ∥}_{2} \\ ⩽ C {∥ e^{- t {| ξ |}^{2}} ((t^{2} {| ξ |}^{2} + t) {| ξ |}^{2 ([N / 2] + j + 1)} + {| ξ |}^{2 ([N / 2] + j)}) χ_{L} (| ξ |) ∥}_{2} + C e^{- δ t} \\ ⩽ C t^{2} {∥ e^{- t {| ξ |}^{2}} {| ξ |}^{2 ([N / 2] + j + 2)} χ_{L} (| ξ |) ∥}_{2} + C t {∥ e^{- t {| ξ |}^{2}} {| ξ |}^{2 ([N / 2] + j + 1)} χ_{L} (| ξ |) ∥}_{2} \\ + {∥ e^{- t {| ξ |}^{2}} {| ξ |}^{2 ([N / 2] + j)} χ_{L} (| ξ |) ∥}_{2} + C e^{- δ t} \\ ⩽ C t^{- n / 4 - [N / 2] - j}, \end{array}$ where we have again used the fact that the supports of $χ_{L}^{'}$ and $χ_{L}^{″}$ do not include the neighborhood of the origin $ξ = 0$ . The estimate (4.26) for $ℓ = 1$ is derived by the interpolation inequality (4.41) again. Thus we have proved (4.26). Due to the previous observation and (4.30), we also have the estimates (4.27)–(4.29). Here we recall the definition of $A_{J} (t)$ ( $J = 0, 1$ ) ((3.2) and (3.4)) and the representation formulas (4.24): $\begin{array}{rcl} A_{0} (t, ξ) = e^{- t {| ξ |}^{2}} (\frac{{(- t)}^{[N / 2] + 1}}{([N / 2] + 1)!} m_{0} (t, ξ) + \sum_{j = 0}^{[N / 2]} \frac{{(- t)}^{j}}{([N / 2] - j + 1)!} m_{0 j} (t, ξ)), \\ A_{1} (t, ξ) = e^{- t {| ξ |}^{2}} (\frac{{(- t)}^{[N / 2] + 1}}{([N / 2] + 1)!} m_{1} (t, ξ) + \sum_{j = 0}^{[N / 2]} \frac{{(- t)}^{j}}{([N / 2] - j + 1)!} m_{1 j} (t, ξ) \\ + \sum_{j = 0}^{[N / 2]} \sum_{k = 0}^{[N / 2] - j} \frac{{(- t)}^{j} α_{j, k}}{([N / 2] - j - k + 1)!} m_{1 j k} (t, ξ)) . \end{array}$ Then by the estimates (4.25) and (4.26), we have $\begin{array}{rclr} {∥ A_{0} (t) ∥}_{{\dot{H}}^{ℓ}} & ⩽ & C t^{[N / 2] + 1} {∥ e^{- t {| ξ |}^{2}} m_{0} (t, ξ) ∥}_{{\dot{H}}^{ℓ}} + C \sum_{j = 0}^{[N / 2]} t^{j} {∥ e^{- t {| ξ |}^{2}} m_{0 j} (t, ξ) ∥}_{{\dot{H}}^{ℓ}} \\ ⩽ & C t^{[N / 2] + 1 - n / 4 - 2 ([N / 2] + 1) + ℓ / 2} + C t^{j - n / 4 - ([N / 2] + 1 + j) + ℓ / 2} \\ ⩽ & C t^{- n / 4 - ([N / 2] + 1) + ℓ / 2} . & (4.42) \end{array}$ Similarly, we use the estimates (4.27)–(4.29) to obtain $\begin{array}{rclr} {∥ A_{1} (t) ∥}_{{\dot{H}}^{ℓ}} & ⩽ & C t^{[N / 2] + 1} {∥ e^{- t {| ξ |}^{2}} m_{1} (t, ξ) ∥}_{{\dot{H}}^{ℓ}} + C \sum_{j = 0}^{[N / 2]} t^{j} {∥ e^{- t {| ξ |}^{2}} m_{1 j} (t, ξ) ∥}_{{\dot{H}}^{ℓ}} \\ + C \sum_{j = 0}^{[N / 2]} \sum_{k = 0}^{[N / 2] - j} t^{j} {∥ e^{- t {| ξ |}^{2}} m_{1 j k} (t, ξ) ∥}_{{\dot{H}}^{ℓ}} \\ ⩽ & C t^{[N / 2] + 1 - n / 4 - 2 ([N / 2] + 1) + ℓ / 2} + C t^{j - n / 4 - ([N / 2] + 1 + j) + ℓ / 2} \\ ⩽ & C t^{- n / 4 - ([N / 2] + 1) + ℓ / 2} . & (4.43) \end{array}$ Then the estimates (4.42), (4.43) and Lemma 2.4 show that $\begin{array}{rcl} M_{\infty} (A_{J} (t)) ⩽ C {∥ A_{J} (t) ∥}_{2}^{1 / 2} {∥ A_{J} (t) ∥}_{{\dot{H}}^{1}}^{1 / 2} ⩽ C t^{- [N / 2] - 1} for n = 1, J = 0, 1, \\ M_{\infty} (A_{J} (t)) ⩽ C {∥ A_{J} (t) ∥}_{2}^{1 - n / 4} {∥ A_{J} (t) ∥}_{{\dot{H}}^{2}}^{n / 4} ⩽ C t^{- [N / 2] - 1} for n = 2, 3, J = 0, 1 . \end{array}$ Therefore we have $M_{q} (A_{J} (t)) ⩽ M_{\infty} (A_{J} (t)) ⩽ C t^{- [N / 2] - 1} for J = 0, 1, 1 ⩽ q ⩽ \infty$ by Lemma 2.3. This means the estimates (4.10) and (4.12) for $t ⩾ 1$ , and the proof of Lemma 4.2 is complete. □
Proof of Proposition 4.1 (continued).
By Lemma 4.2, the estimates (4.6), (4.5), (4.3) and (4.4), we have proved the estimates (4.1) and (4.2), and the proof of Proposition 4.1 is complete. □

5. Weighted estimates

When $n = 1, 2, 3$ , the representation formula of $K_{1} (t) g$ is well known (see Courant and Hilbert [6]) $K_{1} (t) g = \{\begin{matrix} \frac{e^{- t / 2}}{2} \int_{| z | ⩽ t} I_{0} (\frac{\sqrt{t^{2} - {| z |}^{2}}}{2}) g (x + z) d z & (n = 1), \\ \frac{e^{- t / 2}}{2 π} \int_{| z | ⩽ t} \frac{cosh ((\sqrt{t^{2} - {| z |}^{2}}) / 2)}{\sqrt{t^{2} - {| z |}^{2}}} g (x + z) d z & (n = 2), \\ \frac{e^{- t / 2}}{4 π t} \partial_{t} \int_{| z | ⩽ t} I_{0} (\frac{\sqrt{t^{2} - {| z |}^{2}}}{2}) g (x + z) d z & (n = 3), \end{matrix}$ where $I_{ν}$ is the modified Bessel function of order ν defined by $I_{ν} (y) = \sum_{m = 0}^{\infty} \frac{1}{m! Γ (m + ν + 1)} {(\frac{y}{2})}^{2 m + ν} .$ By changing the variable $z = t y$ , we see that $K_{1} (t) g = \{\begin{matrix} \frac{e^{- t / 2} t}{2} \int_{| y | ⩽ 1} I_{0} (\frac{t \sqrt{1 - y^{2}}}{2}) g (x + t y) d y & (n = 1), \\ \frac{e^{- t / 2} t}{2 π} \int_{| y | ⩽ 1} \frac{cosh ((t \sqrt{1 - {| y |}^{2}}) / 2)}{\sqrt{1 - {| y |}^{2}}} g (x + t y) d y & (n = 2), \\ \frac{e^{- t / 2} t^{2}}{4 π} \int_{| y | ⩽ 1} I_{1} (\frac{t \sqrt{1 - {| y |}^{2}}}{2}) g (x + t y) d y \\ + \frac{e^{- t / 2} t}{4 π} \int_{S^{2}} g (x + t y) d σ (y) & (n = 3) . \end{matrix}$ (5.1) For $n = 3$ , such decomposition firstly pointed out by Nishihara [27].

Here, we prepare the point-wise estimate for the modified Bessel function for the proof of $n = 1 and 3$ .

Lemma 5.1.

There exists a constant $C > 0$ such that $| I_{ν} (x) | ⩽ \{\begin{matrix} C x^{ν}, & 0 < x ⩽ 1, \\ C x^{- 1 / 2} e^{x}, & x ⩾ 1, \end{matrix}$ (5.2) for $ν \in Z_{+}$ .

The proof of Lemma 5.1 is immediately seen the representation formulas of the modified Bessel functions. See [1].

Now, we are in a position to show the weighted $L^{1}$ estimates for the evolution operators $K_{0} (t)$ and $K_{1} (t)$ .

Proposition 5.2.

Let $n = 1, 2, 3$ and $m ⩾ 0$ . Then there exists $C > 0$ such that $\begin{array}{rclr} {∥ {| x |}^{m} K_{1} (t) g ∥}_{1} ⩽ \{\begin{matrix} C {∥ {| x |}^{m} g ∥}_{1} + C t^{m / 2} {∥ g ∥}_{1}, & t ⩾ 0, \\ C t ({∥ {| x |}^{m} g ∥}_{1} + t^{m} {∥ g ∥}_{1}), & 0 ⩽ t ⩽ 1, \end{matrix} & (5.3) \\ {∥ {| x |}^{m} K_{0} (t) g ∥}_{1} ⩽ \{\begin{matrix} C {∥ {| x |}^{m} g ∥}_{W^{[n / 2], 1}} + C t^{m / 2} {∥ g ∥}_{W^{[n / 2], 1}}, & t ⩾ 0, \\ C t ({∥ {| x |}^{m} g ∥}_{W^{[n / 2], 1}} + C t^{m / 2} {∥ g ∥}_{W^{[n / 2], 1}}), & 0 ⩽ t ⩽ 1 . \end{matrix} & (5.4) \end{array}$

Remark 5.3.

The estimates (5.3) and (5.4) for $t ⩾ 0$ are obtained in [19] by another method.

Proof of Proposition 5.2.

When $0 ⩽ t ⩽ 1$ , the proof is easy. So we omit the proof here. We only consider the case $t ⩾ 1$ . At first, we show the estimate (5.3). For $n = 1$ , we recall the representation formula of $K_{1} (t) g$ (5.1). Then we use the estimate (5.2) and $- \frac{1}{2} + \frac{\sqrt{1 - {| y |}^{2}}}{2} = \frac{- {| y |}^{2}}{2 (1 + \sqrt{1 - {| y |}^{2}})}$ (5.5) to have $\begin{array}{rcl} | K_{1} (t) g | & ⩽ & C e^{- t / 2} t \int_{| y | ⩽ 1} {(\frac{t \sqrt{1 - y^{2}}}{2})}^{- 1 / 2} e^{(t \sqrt{1 - y^{2}}) / 2} | g (x + t y) | d y \\ = & C t^{1 / 2} \int_{| y | ⩽ 1} \frac{e^{- (t y^{2}) / (2 (1 + \sqrt{1 - y^{2}}))}}{{(1 - y^{2})}^{1 / 4}} | g (x + t y) | d y \\ ⩽ & C t^{1 / 2} \int_{| y | ⩽ 1} \frac{e^{- C t y^{2}}}{{(1 - | y |)}^{1 / 4}} | g (x + t y) | d y . \end{array}$ Therefore the Fubini theorem and the estimate (2.5) show that $\begin{array}{rcl} {∥ {| x |}^{m} K_{1} (t) g ∥}_{L_{x}^{1}} & ⩽ & C t^{1 / 2} \int_{| y | ⩽ 1} \int_{R} ({| x + t y |}^{m} + t^{m} {| y |}^{m}) \frac{e^{- C t y^{2}}}{{(1 - | y |)}^{1 / 4}} | g (x + t y) | d x d y \\ ⩽ & C t^{1 / 2} \int_{| y | ⩽ 1} \frac{e^{- C t y^{2}}}{{(1 - | y |)}^{1 / 4}} d y ∥ g ∥_{L_{m}^{1}} + C t^{m + 1 / 2} \int_{| y | ⩽ 1} \frac{{| y |}^{m} e^{- C t y^{2}}}{{(1 - | y |)}^{1 / 4}} d y {∥ g ∥}_{1} \\ ⩽ & C {∥ g ∥}_{L_{m}^{1}} + C t^{m / 2} {∥ g ∥}_{1} . \end{array}$ Thus, we have the estimate (5.3) for $n = 1$ .

When $n = 2$ , by (5.1) and (5.5), we see that $\begin{array}{rcl} K_{1} (t) g & = & \frac{e^{- t / 2} t}{2 π} \int_{| y | ⩽ 1} \frac{e^{(t \sqrt{1 - {| y |}^{2}}) / 2} + e^{(- t \sqrt{1 - {| y |}^{2}}) / 2}}{2 \sqrt{1 - {| y |}^{2}}} g (x + t y) d y \\ = & \frac{t}{4 π} \int_{| y | ⩽ 1} (\frac{e^{- (t {| y |}^{2}) / (2 (1 + \sqrt{1 - {| y |}^{2}}))} + e^{- t / 2 - (t \sqrt{1 - {| y |}^{2}}) / 2}}{\sqrt{1 - {| y |}^{2}}}) g (x + t y) d y . \end{array}$ Therefore the Fubini theorem and (2.5) again yield $\begin{array}{rcl} {∥ {| x |}^{m} K_{1} (t) g ∥}_{L_{x}^{1}} \\ ⩽ C t \int_{| y | ⩽ 1} \int_{R^{2}} (| x + t y |^{m} + t^{m} {| y |}^{m}) \frac{e^{- (t {| y |}^{2}) / (2 (1 + \sqrt{1 - {| y |}^{2}}))}}{\sqrt{1 - {| y |}^{2}}} | g (x + t y) | d x d y \\ + C e^{- t / 2} t \int_{| y | ⩽ 1} \int_{R^{2}} ({| x + t y |}^{m} + t^{m} {| y |}^{m}) \frac{| g (x + t y) |}{\sqrt{1 - {| y |}^{2}}} d x d y \\ ⩽ C t \int_{| y | ⩽ 1} \frac{e^{- C t {| y |}^{2}}}{\sqrt{1 - {| y |}^{2}}} d y {∥ g ∥}_{L_{m}^{1}} + C t^{m + 1} \int_{| y | ⩽ 1} \frac{{| y |}^{m} e^{- C t {| y |}^{2}}}{\sqrt{1 - {| y |}^{2}}} d y {∥ g ∥}_{1} \\ + C e^{- t / 2} t \int_{| y | ⩽ 1} \frac{1}{\sqrt{1 - | y |^{2}}} d y ({∥ g ∥}_{L_{m}^{1}} + t^{m} {∥ g ∥}_{1}) \\ ⩽ C {∥ g ∥}_{L_{m}^{1}} + C t^{m / 2} {∥ g ∥}_{1} + C e^{- t / 2} t ({∥ g ∥}_{L_{m}^{1}} + t^{m} {∥ g ∥}_{1}) \\ ⩽ C {∥ g ∥}_{L_{m}^{1}} + C t^{m / 2} {∥ g ∥}_{1}, \end{array}$ where we have used the fact that $\int_{| y | ⩽ 1} \frac{1}{\sqrt{1 - {| y |}^{2}}} d y ⩽ C$ for $n = 2$ . Namely, we proved the estimate (5.3) for $n = 2$ .

For the proof of $n = 3$ , we recall the decomposition of $K_{1} (t) g$ (5.1). According to [27], we define $J_{1} (t) g$ and $W_{1} (t) g$ by $\begin{array}{rclr} J_{1} (t) g : = \frac{e^{- t / 2} t^{2}}{2} \int_{| y | ⩽ 1} I_{1} (\frac{t \sqrt{1 - {| y |}^{2}}}{2}) \frac{g (x + t y)}{\sqrt{1 - {| y |}^{2}}} d y, & (5.6) \\ W_{1} (t) g : = \frac{e^{- t / 2} t}{4 π} \int_{S^{2}} g (x + t y) d σ (y) . & (5.7) \end{array}$ By the estimate (5.2), we see that $\begin{array}{rcl} | J_{1} (t) g | & ⩽ & C e^{- t / 2} t^{2} \int_{| y | ⩽ 1} \frac{e^{(t \sqrt{1 - {| y |}^{2}}) / 2}}{{(t \sqrt{1 - {| y |}^{2}})}^{1 / 2}} \frac{| g (x + t y) |}{\sqrt{1 - | y |^{2}}} d y \\ = & C t^{3 / 2} \int_{| y | ⩽ 1} e^{- (t {| y |}^{2}) / (2 (1 + \sqrt{1 - {| y |}^{2}}))} \frac{| g (x + t y) |}{{(1 - {| y |}^{2})}^{3 / 4}} d y \\ ⩽ & C t^{3 / 2} \int_{| y | ⩽ 1} e^{- C t {| y |}^{2}} \frac{| g (x + t y) |}{{(1 - {| y |}^{2})}^{3 / 4}} d y . \end{array}$ Again, we apply the Fubini theorem and (2.5) to have $\begin{array}{rclr} {∥ {| x |}^{m} J_{1} (t) g ∥}_{L_{x}^{1}} & ⩽ & C t^{3 / 2} \int_{| y | ⩽ 1} \int_{R^{3}} ({| x + t y |}^{m} + t^{m} {| y |}^{m}) \frac{e^{- C t {| y |}^{2}}}{{(1 - {| y |}^{2})}^{3 / 4}} | g (x + t y) | d x d y \\ ⩽ & C t^{3 / 2} \int_{| y | ⩽ 1} \frac{e^{- C t {| y |}^{2}}}{{(1 - {| y |}^{2})}^{3 / 4}} d y {∥ g ∥}_{L_{m}^{1}} \\ + C t^{m + 3 / 2} \int_{| y | ⩽ 1} \frac{{| y |}^{m} e^{- C t {| y |}^{2}}}{{(1 - {| y |}^{2})}^{3 / 4}} d y {∥ g ∥}_{1} \\ ⩽ & C {∥ g ∥}_{L_{m}^{1}} + C t^{m / 2} {∥ g ∥}_{1} . & (5.8) \end{array}$ For the term $W_{1} (t) g$ , the Fubini theorem yields $\begin{array}{rclr} {∥ {| x |}^{m} W_{1} (t) g ∥}_{L_{x}^{1}} & ⩽ & C e^{- t / 2} t \int_{S^{2}} \int_{R_{x}^{3}} ({| x + t y |}^{m} + t^{m} {| y |}^{m}) | g (x + t y) | d x d σ (y) \\ ⩽ & C e^{- t / 2} t ({∥ g ∥}_{L_{m}^{1}} + t^{m} {∥ g ∥}_{1}) . & (5.9) \end{array}$ Then, combining the estimates (5.8) and (5.9) gives $\begin{array}{rcl} {∥ {| x |}^{m} K_{1} (t) g ∥}_{L_{x}^{1}} & ⩽ & {∥ {| x |}^{m} J_{1} (t) g ∥}_{L_{x}^{1}} + {∥ {| x |}^{m} W_{1} (t) g ∥}_{L_{x}^{1}} \\ ⩽ & C {∥ g ∥}_{L_{m}^{1}} + C t^{m / 2} {∥ g ∥}_{1} + C e^{- t / 2} t ({∥ g ∥}_{L_{m}^{1}} + t^{m} {∥ g ∥}_{1}) \\ ⩽ & C {∥ g ∥}_{L_{m}^{1}} + C t^{m / 2} {∥ g ∥}_{1} \end{array}$ and we have proved the estimate (5.3) for $n = 3$ . Thus we have (5.3) for $n = 1, 2, 3$ . Finally we prove (5.4). For this purpose we note that $K_{0} (t) g = \partial_{t} K_{1} (t) g + \frac{1}{2} K_{1} (t) g .$ (5.10) When $n = 1 and 2$ , the weight estimate (5.4) is shown by the same method as in (5.3). Indeed, for $n = 1$ , Marcati and Nishihara [23] proved that $\begin{array}{rcl} K_{0} (t) g & = & \partial_{t} K_{1} (t) g + \frac{1}{2} K_{1} (t) g \\ = & \frac{1}{2} K_{1} (t) g + \frac{e^{- t / 2} t}{4} \int_{| z | ⩽ t} I_{1} (\frac{\sqrt{t^{2} - z^{2}}}{2}) \frac{g (x - z)}{\sqrt{t^{2} - z^{2}}} d z \\ + e^{- t / 2} (g (x - t) + g (x + t)), \end{array}$ where $I_{0}^{'} (z) = I_{1} (z)$ . Observing that $\frac{e^{- t / 2} t}{4} \int_{| z | ⩽ t} I_{1} (\frac{\sqrt{t^{2} - z^{2}}}{2}) \frac{g (x - z)}{\sqrt{t^{2} - z^{2}}} d z = \frac{e^{- t / 2} t}{4} \int_{| y | ⩽ 1} I_{1} (\frac{t \sqrt{1 - y^{2}}}{2}) \frac{g (x + t y)}{\sqrt{1 - y^{2}}} d y,$ we use the estimates (5.3), (5.2) and (5.5) to see that $\begin{array}{rcl} | K_{0} (t) g | & ⩽ & C | K_{1} (t) g | + C e^{- t / 2} t^{1 / 2} \int_{| y | ⩽ 1} e^{(t \sqrt{1 - y^{2}}) / 2} \frac{g (x + t y)}{{(1 - y^{2})}^{3 / 4}} d y \\ + e^{- t / 2} (| g (x - t) | + | g (x + t) |) . \end{array}$

Therefore we apply the Fubini theorem, the estimates (2.5) and (5.3) to have $\begin{array}{rclr} {∥ {| x |}^{m} K_{0} (t) g ∥}_{L_{x}^{1}} \\ ⩽ {∥ {| x |}^{m} K_{1} (t) g ∥}_{1} + C t^{1 / 2} \int_{R} ({| x + t y |}^{m} + {| t y |}^{m}) \int_{| y | ⩽ 1} e^{- C t y^{2}} \frac{| g (x + t y) |}{{(1 - y^{2})}^{3 / 4}} d y d x \\ + C e^{- δ t} {∥ g ∥}_{L_{m}^{1}} \\ ⩽ {∥ {| x |}^{m} K_{1} (t) g ∥}_{1} + C {∥ g ∥}_{L_{m}^{1}} + C t^{m / 2} {∥ g ∥}_{1} + C e^{- δ t} {∥ g ∥}_{L_{m}^{1}} \\ ⩽ {∥ {| x |}^{m} K_{1} (t) g ∥}_{1} + C {∥ g ∥}_{L_{m}^{1}} + C t^{m / 2} {∥ g ∥}_{1} . & (5.11) \end{array}$ For $t ⩾ 1$ , applying the estimates (5.3) and (5.11) directly yields the desired estimate (5.4) for $n = 1$ .

When $n = 2$ , we have $\begin{array}{rcl} K_{0} (t) g & = & \partial_{t} K_{1} (t) g + \frac{1}{2} K_{1} (t) g \\ = & t^{- 1} K_{1} (t) g + \frac{e^{- t / 2 t} t}{2 π} \int_{| y | ⩽ 1} \partial_{t} (\frac{cosh (t (\sqrt{1 - {| y |}^{2}}) / 2)}{\sqrt{1 - {| y |}^{2}}} g (x + t y)) d y \\ = & t^{- 1} K_{1} (t) g + \frac{e^{- t / 2 t} t}{2 π} \int_{| y | ⩽ 1} sinh (t \frac{\sqrt{1 - {| y |}^{2}}}{2}) g (x + t y) d y \\ + \frac{e^{- t / 2 t} t}{2 π} \int_{| y | ⩽ 1} \frac{cosh (t (\sqrt{1 - {| y |}^{2}}) / 2)}{\sqrt{1 - {| y |}^{2}}} \nabla g (x + t y) \cdot y d y \end{array}$ by (5.1). Therefore, as in the proof of (5.11), we arrive at the estimate $\begin{array}{rclr} {∥ {| x |}^{m} K_{0} (t) g ∥}_{L_{x}^{1}} \\ ⩽ C t^{- 1} {{∥ {| x |}^{m} K_{1} (t) g ∥}_{1} + C (1 + t)}^{m / 2} {∥ \nabla g ∥}_{1} + C {∥ \nabla g ∥}_{L_{m}^{1}} + C {∥ g ∥}_{L_{m}^{1}} + C t^{m / 2} {∥ g ∥}_{1} . & (5.12) \end{array}$ For $t ⩾ 1$ , the estimate (5.12) means the desired estimate (5.4).

To show the case $n = 3$ , we observe that $\begin{array}{rcl} K_{0} (t) g & = & \partial_{t} K_{1} (t) g + \frac{1}{2} K_{1} (t) g = \partial_{t} J_{1} (t) g + \partial_{t} W_{1} (t) g \\ = & 2 t^{- 1} J_{1} (t) g + \frac{e^{- t / 2} t^{2}}{2} \int_{| y | ⩽ 1} \partial_{t} (I_{1} (\frac{t \sqrt{1 - | y |^{2}}}{2}) \frac{g (x + t y)}{\sqrt{1 - {| y |}^{2}}}) d y + \partial_{t} W_{1} (t) g \end{array}$ by (5.1), (5.6) and (5.7). For the second term in the right-hand side, we apply the well-known formula (see Nishihara [27]) $I_{1}^{'} (y) = I_{0} (y) - \frac{1}{y} I_{1} (y)$ to see that $\begin{array}{rcl} \frac{e^{- t / 2} t^{2}}{2} \int_{| y | ⩽ 1} \partial_{t} (I_{1} (\frac{t \sqrt{1 - {| y |}^{2}}}{2}) \frac{g (x + t y)}{\sqrt{1 - {| y |}^{2}}}) d y \\ = \frac{e^{- t / 2} t^{2}}{2} \int_{| y | ⩽ 1} \partial_{t} (I_{1} (\frac{t \sqrt{1 - {| y |}^{2}}}{2})) \frac{g (x + t y)}{\sqrt{1 - {| y |}^{2}}} d y \\ + \frac{e^{- t / 2} t^{2}}{2} \int_{| y | ⩽ 1} I_{1} (\frac{t \sqrt{1 - {| y |}^{2}}}{2}) \frac{\nabla g (x + t y) \cdot y}{\sqrt{1 - {| y |}^{2}}} d y \\ = \frac{e^{- t / 2} t^{2}}{4} \int_{| y | ⩽ 1} I_{0} (\frac{t \sqrt{1 - {| y |}^{2}}}{2}) g (x + t y) d y + t^{- 1} J_{1} (t) g \\ + \frac{e^{- t / 2} t^{2}}{2} \int_{| y | ⩽ 1} I_{1} (\frac{t \sqrt{1 - {| y |}^{2}}}{2}) \frac{\nabla g (x + t y) \cdot y}{\sqrt{1 - {| y |}^{2}}} d y \end{array}$ and $\begin{array}{rcl} | \frac{e^{- t / 2} t^{2}}{2} \int_{| z | ⩽ 1} \partial_{t} (I_{1} (\frac{t \sqrt{1 - {| y |}^{2}}}{2}) \frac{g (x + t y)}{\sqrt{1 - {| y |}^{2}}}) d y | \\ ⩽ C e^{- t / 2} t^{2} \int_{| y | ⩽ 1} {(\frac{t \sqrt{1 - {| y |}^{2}}}{2})}^{- 1} e^{(t \sqrt{1 - {| y |}^{2}}) / 2} | g (x + t y) | d y + t^{- 1} | J_{1} (t) g | + | K_{1} (t) | \nabla g | | \\ ⩽ C t^{3 / 2} \int_{| y | ⩽ 1} \frac{e^{(- t {| y |}^{2}) / (2 (1 + \sqrt{1 - {| y |}^{2}}))}}{{(\sqrt{1 - {| y |}^{2}})}^{1 / 2}} | g (x + t y) | d y + t^{- 1} | J_{1} (t) g | + | K_{1} (t) | \nabla g | | . \end{array}$ Therefore we obtain the point-wise estimate for $K_{0} (t) g$ $\begin{array}{rclr} | K_{0} (t) g | & ⩽ & C t^{- 1} | J_{1} (t) g | + C t^{3 / 2} \int_{| y | ⩽ 1} \frac{e^{(- t | y |^{2}) / (2 (1 + \sqrt{1 - | y |^{2}}))}}{{(\sqrt{1 - {| y |}^{2}})}^{1 / 2}} | g (x + t y) | d y \\ + | K_{1} (t) | \nabla g | | + C | W_{1} (t) g | + C | W_{1} (t) | \nabla g | | . & (5.13) \end{array}$ In (5.13), we only need to estimate the second term in the right-hand side $C t^{3 / 2} \int_{| y | ⩽ 1} \frac{e^{(- t {| y |}^{2}) / (2 (1 + \sqrt{1 - {| y |}^{2}}))}}{{(\sqrt{1 - {| y |}^{2}})}^{1 / 2}} | g (x + t y) | d y .$ Indeed, by the estimates (5.3) and (5.9), we have $\begin{array}{rcl} {∥ {| x |}^{m} K_{0} (t) g ∥}_{L_{x}^{1}} \\ ⩽ t^{- 1} {∥ {| x |}^{m} J_{1} (t) g ∥}_{1} + C t^{3 / 2} \int_{| y | ⩽ 1} \int_{R^{3}} ({| x + t y |}^{m} + t^{m} {| y |}^{m}) \frac{e^{- C t {| y |}^{2}}}{{(\sqrt{1 - {| y |}^{2}})}^{1 / 2}} | g (x + t y) | d x d y \\ + {∥ {| x |}^{m} K_{1} (t) | \nabla g | ∥}_{1} + C {∥ {| x |}^{m} W_{1} (t) g ∥}_{1} + C {∥ {| x |}^{m} W_{1} (t) | \nabla g | ∥}_{1} \\ ⩽ t^{- 1} {∥ {| x |}^{m} J_{1} (t) g ∥}_{1} + C t^{3 / 2} \int_{| y | ⩽ 1} \frac{e^{- C t {| y |}^{2}}}{{(\sqrt{1 - {| y |}^{2}})}^{1 / 2}} d y {∥ g ∥}_{1} \\ + C t^{m} \int_{| y | ⩽ 1} \frac{e^{- C t {| y |}^{2}} {| y |}^{m}}{{(\sqrt{1 - {| y |}^{2}})}^{1 / 2}} d y {∥ g ∥}_{1} + C t^{m / 2} {∥ g ∥}_{W^{1, 1}} + C {∥ {| x |}^{m} g ∥}_{W_{m}^{1, 1}} . \end{array}$ As in the proof of (2.5), we see that $\int_{| y | ⩽ 1} \frac{e^{- C t {| y |}^{2}} {| y |}^{m}}{{(\sqrt{1 - {| y |}^{2}})}^{1 / 2}} d y ⩽ C t^{- m / 2 - 3 / 2}$ (5.14) for $n = 3$ and $m ⩾ 0$ . Then we obtain the estimate: ${∥ | x |^{m} K_{0} (t) g ∥}_{L_{x}^{1}} ⩽ t^{- 1} {∥ {| x |}^{m} J_{1} (t) g ∥}_{1} + C {∥ g ∥}_{1} + C t^{m / 2} {∥ g ∥}_{W^{1, 1}} + C {∥ {| x |}^{m} g ∥}_{W_{m}^{1, 1}} .$ (5.15) For $t ⩾ 1$ , the combination of the estimates (5.8) and (5.14) implies the estimate (5.4) for $n = 3$ . Thus, we complete the proof of the estimate (5.4) for $n = 1, 2, 3$ . □

Summing up Theorem 1.1 and Proposition 5.2, we have the approximation of the solution to (1.1) by the Gauss kernel.

Proof of Corollary 1.2.

By Lemma 2.1 and Proposition 5.2, we easily see that the solution $u (t)$ is in the class $C ([0, \infty); L_{N}^{1} \cap L^{\infty})$ . Moreover, the estimates (1.7) and (2.3) yield that $\begin{array}{rcl} ∥ u (t) - \frac{1}{2} \sum_{j = 0}^{[N / 2]} \sum_{k = 0}^{[N / 2] - j} \sum_{| γ | ⩽ N - 2 (j + k)} α_{j, k} μ_{0 γ} {(- t)}^{j} {(- Δ)}^{2 j + k} \partial_{x}^{γ} G_{t} \\ {- \sum_{j = 0}^{[N / 2]} \sum_{k = 0}^{[N / 2] - j} \sum_{ℓ = 0}^{[N / 2] - j - k} \sum_{| γ | ⩽ N - 2 (j + k + ℓ)} α_{j, k} β_{ℓ} (\frac{1}{2} μ_{0 γ} + μ_{1 γ}) {(- t)}^{j} {(- Δ)}^{2 j + k + ℓ} \partial_{x}^{γ} G_{t} ∥}_{q} \\ ⩽ ∥ u (t) - \frac{1}{2} \sum_{j = 0}^{[N / 2]} \sum_{k = 0}^{[N / 2] - j} α_{j, k} {(- t)}^{j} {(- Δ)}^{2 j + k} e^{t Δ} g_{0} \\ {- \sum_{j = 0}^{[N / 2]} \sum_{k = 0}^{[N / 2] - j} \sum_{ℓ = 0}^{[N / 2] - j - k} α_{j, k} β_{ℓ} {(- t)}^{j} {(- Δ)}^{2 j + k + ℓ} e^{t Δ} (\frac{1}{2} g_{0} + g_{1}) ∥}_{q} \\ + {∥ \frac{1}{2} \sum_{j = 0}^{[N / 2]} \sum_{k = 0}^{[N / 2] - j} α_{j, k} {(- t)}^{j} {(- Δ)}^{2 j + k} (e^{t Δ} g_{0} - \sum_{| γ | ⩽ N - 2 (j + k)} μ_{0 γ} \partial_{x}^{γ} G_{t}) ∥}_{q} \\ + ∥ \sum_{j = 0}^{[N / 2]} \sum_{k = 0}^{[N / 2] - j} \sum_{ℓ = 0}^{[N / 2] - j - k} α_{j, k} β_{ℓ} {(- t)}^{j} {(- Δ)}^{2 j + k + ℓ} \\ {\times (e^{t Δ} (\frac{1}{2} g_{0} + g_{1}) - \sum_{| γ | ⩽ N - 2 (j + k + ℓ)} (\frac{1}{2} μ_{0 γ} + μ_{1 γ}) \partial_{x}^{γ} G_{t}) ∥}_{q} \\ ⩽ C t^{- n / 2 (1 - 1 / q) - ([N / 2] + 1)} ({∥ g_{0} ∥}_{W^{[n / 2], 1} \cap W^{[n / 2], \infty}} + {∥ g_{1} ∥}_{L^{1} \cap L^{\infty}}) + o (t^{- (n / 2) (1 - 1 / q) - [N / 2]}) \end{array}$ as $t \to \infty$ . Then we have the estimate (1.8). The proof of Corollary 1.2 is complete. □

Footnotes

Acknowledgements

The author would like to express sincere gratitude to Professor Takayoshi Ogawa for valuable discussion. The author is grateful to Professor Shuji Yoshikawa for his many encouragement and helpful comment. The author is also grateful to Professor Tatsuki Kawakami for his valuable conversation. The author would like to thank the referee for several helpful suggestions.

References

[1]

Abramowitz and

I.A.

Stegun (eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications Inc., New York, 1992. (Reprint of the 1972 edition.)

[2]

Bellout and

Friedman, Blow-up estimates for a nonlinear hyperbolic heat equation, SIAM J. Math. Anal. 20 (1989), 354–366.

[3]

Bergh and

Löfström, Interpolation Spaces, Grundlehren der mathematischen Wissenschaften, Vol. 223, Springer, Berlin, New York, 1976.

[4]

Brenner,

Thomée and

Wahlbin, Besov Spaces and Applications to Difference Methods for Initial Value Problems, Lecture Notes in Mathematics, Vol. 434, Springer, Berlin, New York, 1975.

[5]

Chill and

Haraux, An optimal estimate for the difference of solutions of two abstract evolution equations, J. Differential Equations 193 (2003), 385–395.

[6]

Courant and

Hilbert, Methods of Mathematical Physics, Vol. II: Partial Differential Equations, Interscience Publishers (a division of John Wiley & Sons), New York, London, 1962.

[7]

Dan and

Shibata, On a local energy decay of solutions of a dissipative wave equation, Funkcial. Ekvac. 38 (1995), 545–568.

[8]

Gallay and

Raugel, Scaling variables and asymptotic expansions in damped wave equations, J. Differential Equations 150 (1998), 42–97.

[9]

M.-H.

Giga,

Giga and

Saal, Nonlinear Partial Differential Equations. Asymptotic Behavior of Solutions and Self-Similar Solutions, Progress in Nonlinear Differential Equations and Their Applications, Vol. 79, Birkhäuser, Boston, MA, 2010.

10.

[10]

Hayashi,

Kaikina and

P.I.

Naumkin, Damped wave equation with super critical nonlinearities, Differential Integral Equations 17 (2004), 637–652.

11.

[11]

Hayashi,

Kaikina and

P.I.

Naumkin, Damped wave equation with a critical nonlinearity, Trans. Amer. Math. Soc. 358 (2006), 1165–1185.

12.

[12]

Hirosawa and

Wirth,

C^{m}

-theory of damped wave equations with stabilisation, J. Math. Anal. Appl. 343 (2008), 1022–1035.

13.

[13]

Hosono and

Ogawa, Large time behavior and

L^{p}

–

L^{q}

estimate of 2-dimensional nonlinear damped wave equations, J. Differential Equations 203 (2004), 82–118.

14.

[14]

Ikehata,

Miyaoka and

Nakatake, Decay estimates of solutions for dissipative wave equations in

R^{n}

with lower power nonlinearities, J. Math. Soc. Japan 56 (2004), 365–373.

15.

[15]

Ikehata and

Tanizawa, Global existence of solutions for semilinear wave equations in

R^{N}

with non-compactly supported initial data, Nonlinear Anal. T.M.A. 61 (2005), 1189–1208.

16.

[16]

Ishige,

Ishiwata and

Kawakami, The decay of the solutions for the heat equation with a potential, Indiana Univ. Math. J. 58 (2009), 2673–2707.

17.

[17]

Ishige and

Kawakami, Refined asymptotic profiles for a semilinear heat equation, Math. Ann. 353 (2012), 161–192.

18.

[18]

Karch, Selfsimilar profiles in large time asymptotics of solutions to damped wave equations, Studia Math. 143 (2000), 175–197.

19.

[19]

Kawakami and

Ueda, Asymptotic profiles to the solutions for a nonlinear damped wave equations, Differential Integral Equations 26 (2013), 781–814.

20.

[20]

Kawashima,

Nakao and

Ono, On the decay property of solutions to the Cauchy problem of the semilinear wave equation with a dissipative term, J. Math. Soc. Japan 47 (1995), 617–653.

21.

[21]

T.-T.

Li, Nonlinear heat conduction with finite speed of propagation, in: China–Japan Symposium on Reaction–Diffusion Equations and Their Applications and Computational Aspects, Shanghai, 1994, World Sci. Publ., River Edge, NJ, 1997, pp. 81–91.

22.

[22]

T.-T.

Li and

Zhou, Breakdown of the solutions to

□ u + u_{t} = | u |^{1 + α}

, Discrete Contin. Dynam. Systems 1 (1995), 503–520.

23.

[23]

Marcati and

Nishihara, The

L^{p}

–

L^{q}

estimates of solutions to one-dimensional damped wave equations and their application to compressible flow through porous media, J. Differential Equations 191 (2003), 445–469.

24.

[24]

Matsumura, On the asymptotic behavior of solutions of semilinear wave equations, Publ. Res. Inst. Sci. Kyoto Univ. 12 (1976), 169–189.

25.

[25]

Nakao and

Ono, Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations, Math. Z. 214 (1993), 325–342.

26.

[26]

Narazaki,

L^{p}

–

L^{q}

estimates for damped wave equations and their applications to semi-linear problem, J. Math. Soc. Japan 56 (2004), 585–626.

27.

[27]

Nishihara,

L^{p}

–

L^{q}

estimates of solutions to the damped wave equation in 3-dimensional space and their application, Math. Z. 244 (2003), 631–649.

28.

[28]

Ono, Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations, Discrete Contin. Dynam. Systems 9 (2003), 651–662.

29.

[29]

Orive,

Zuazua and

A.F.

Pazoto, Asymptotic expansion for damped wave equations with periodic coefficients, Math. Methods Appl. Sci. 11 (2001), 1285–1310.

30.

[30]

Radu,

Todorova and

Yordanov, Higher order energy decay rates for damped wave equations with variable coefficients, Discrete Contin. Dyn. Syst. Ser. S 2 (2009), 609–629.

31.

[31]

Takeda, Large time behavior of solutions for a nonlinear damped wave equation, submitted for publication.

32.

[32]

Takeda and

Yoshikawa, Asymptotic profiles of solutions for the isothermal Falk–Konopka system of shape memory alloys with weak damping, Asymptotic Anal. 81 (2013), 331–372.

33.

[33]

Todorova and

Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001), 464–489.

34.

[34]

Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris 333 (2001), 109–114.