Gevrey Properties and Summability of Formal Power Series Solutions of Some Inhomogeneous Linear Cauchy-Goursat Problems

Abstract

In this article, we investigate the Gevrey and summability properties of the formal power series solutions of some inhomogeneous linear Cauchy-Goursat problems with analytic coefficients in a neighborhood of $(0,0)\in \mathbb {C}^{2}$. In particular, we give necessary and sufficient conditions under which these solutions are convergent or are k-summable, for a convenient positive rational number k, in a given direction.

Appendix:: Gevrey asymptotic

In this IAppendix, we present various results of the general theory of the Gevrey asymptotic expansions in the framework of the formal power series in $\mathcal {O}(D_{\rho _2})[[t]]$.

1.1 s-Gevrey asymptotic

Still considering t as the variable and x as a parameter, one extends, in the similar way as the s-Gevrey formal series (see Definition 2), the classical notion of Gevrey asymptotic to a formal series in $\mathbb {C}[[t]]$ to the one of Gevrey asymptotic to a formal series in $\mathcal {O}(D_{\rho _{2}})[[t]]$ in requiring similar conditions, the estimates being however uniform with respect to x.

Definition 5 (s-Gevrey asymptotic)

Let $s\geqslant 0$ and Σ be an open sector with vertex $0\in \mathbb {C}$. A function u(t, x) holomorphic on a domain Σ × D_ρ for some ρ > 0 is said to be Gevrey asymptotic of order s (in short, s-Gevrey asymptotic) to a formal series$\displaystyle {\sum }_{j\geqslant 0}u_{j,\ast }(x)\frac {t^{j}}{j!}\in \mathcal {O}(D_{\rho _{2}})[[t]]$on Σ if there exists 0 < r₂ < min(ρ, ρ₂) such that, for any proper subsector Σ^′ ⋐ Σ, there exist two positive constants C > 0 and K > 0 such that, for all $J\geqslant 1$ and all t ∈Σ^′:

$$ \sup_{|x|\leqslant r_{2}}\left\vert u(t,x)-{\sum}_{j = 0}^{J-1}u_{j,\ast}(x)\frac{t^{j}}{j!}\right\vert\leqslant CK^{J}{\Gamma}(1+sJ \null)\left\vert t\right\vert^{J}. $$

(1)

A series which is the s-Gevrey asymptotic expansion of a function is said to be an s-Gevrey asymptotic series on Σ.

Remark 12

If any exists, the s-Gevrey asymptotic series is unique.

Proposition 7

Let$s\geqslant 0$be.Then, a s-Gevrey asymptotic series on a sectorΣ is a s-Gevrey series.

Proof

Let $\widetilde {u}(t,x)=\displaystyle {\sum }_{j\geqslant 0}u_{j,\ast }(x)\frac {t^{j}}{j!}\in \mathcal {O}(D_{\rho _{2}})[[t]]$ be a s-Gevrey asymptotic series of a function u(t, x) on Σ. We want to prove that there exist positive constants $0<r_{2}^{\prime \prime }<\rho _{2}$, C^″ > 0 and K^″ > 0 such that, for all $J\geqslant 0$,

$$ \sup_{\left\vert x\right\vert\leqslant r_{2}^{\prime\prime}}\left\vert u_{J,\ast}(x)\right\vert\leqslant C^{\prime\prime}K^{{\prime\prime}J}{\Gamma}(1+(s + 1)J \null). $$

(2)

Let r₂ > 0 be as in Definition 5 and let us choose Σ^′ ⋐ Σ a proper subsector of Σ. For any $J\geqslant 1$, we derive from condition (1) applied twice to the relation

$$u_{J,\ast}(x)\frac{t^{J}}{J!}=\left( u(t,x)-{\sum}_{j = 0}^{J-1}u_{j,\ast}(x)\frac{t^{j}}{j!}\right)-\left( u(t,x)-{\sum}_{j = 0}^{J}u_{j,\ast}(x)\frac{t^{j}}{j!}\right) $$

the following inequality

$$\sup_{\left\vert x\right\vert\leqslant r_{2}}\left\vert u_{J,\ast}(x)\right\vert\leqslant CK^{J}{\Gamma}(1+sJ \null)J!+CK^{J + 1}{\Gamma}(1+s(J + 1) \null)J!R, $$

where R > 0 denotes the radius of Σ^′. Applying then the relation between the Gamma and the Beta functions to Γ(1 + sJ)J! = Γ(1 + sJ)Γ(1 + J), we get

$${\Gamma}(1+sJ \null)J!={\Gamma}(2+(s + 1)J){{\int}_{0}^{1}}t^{sJ}(1-t)^{J}dt\leqslant{\Gamma}(2+(s + 1)J); $$

hence, the inequalities

$${\Gamma}(1+sJ \null)J!\leqslant(1+(s + 1)J){\Gamma}(1+(s + 1)J \null)\leqslant e\left( e^{s + 1}\right)^{J}{\Gamma}(1+(s + 1)J \null). $$

In the same way, and using besides the increase of the Gamma function on [2, + ∞[, we have

$${\Gamma}(1+s(J + 1))J!\leqslant{\Gamma}(2+(s + 1)J+s)\leqslant{\Gamma}(2+(s + 1)J+S), $$

where S is an integer $\geqslant s$; hence,

$${\Gamma}(1+s(J + 1))J!\leqslant{\Gamma}(1+(s + 1)J \null){\prod}_{\ell= 1}^{S + 1}((s + 1)J+\ell)\leqslant AB^{J}{\Gamma}(1+(s + 1)J \null) $$

with convenient constants A, B > 0 independent of J. Consequently, there exist C^′, K^′ > 0 such that the following inequalities

$$\sup_{\left\vert x\right\vert\leqslant r_{2}}\left\vert u_{J,\ast}(x)\right\vert\leqslant C^{\prime}K^{{\prime}J}{\Gamma}(1+(s + 1)J \null) $$

hold for all $J\geqslant 1$. Condition (2) follows then by choosing

$$r_{2}^{\prime\prime}=r_{2}\text{, }C^{\prime\prime}=\max\left( C^{\prime},\sup_{\left\vert x\right\vert\leqslant r_{2}}\left\vert u_{0,\ast}(x)\right\vert\right)\text{ and }K^{\prime\prime}=K^{\prime}.$$

This ends the proof. □

Following Proposition 8 gives us a characterization of the s-Gevrey asymptotic in terms of conditions on the successive derivatives ${\partial _{t}^{J}} u$ of the function u with respect to t.

Proposition 8

Let$s\geqslant 0$andΣ be an open sectorwith vertex$0\in \mathbb {C}$. Then, afunctionu(t, x) holomorphicon a domain Σ × D_ρfor someρ > 0 is s-Gevrey asymptoticto a formal series$\widetilde {u}(t,x)=\displaystyle {\sum }_{j\geqslant 0}u_{j,\ast }(x)\frac {t^{j}}{j!}\in \mathcal {O}(D_{\rho _{2}})[[t]]$onΣ if and onlyif there exists 0 < r₂ < min(ρ, ρ₂) such that

• 1. For any |x| ≤ r₂,the mapt↦u(t, x) has$\widetilde {u}(t,x)$asTaylor series at 0 on Σ,

• 2. For any proper subsector Σ^′ ⋐ Σ,there exist two positive constantsC > 0 andK > 0 such that,for all$J\geqslant 0$and allt ∈Σ^′,

  $$\sup_{\left\vert x\right\vert\leqslant r_{2}}\left\vert{\partial_{t}^{J}}u(t,x)\right\vert\leqslant CK^{J}{\Gamma}(1+(s + 1)J \null). $$

Proof

⊲ Necessary condition. Let us suppose that u(t, x) is s-Gevrey asymptotic to $\widetilde {u}(t,x)$ on Σ and let us prove Conditions 1 and 2 of Proposition 8.

Due to Definition 5, Condition 1 is straightforward. To prove Condition 2, we consider 0 < r₂ < min(ρ, ρ₂) as in Definition 5 and a proper subsector Σ^′ ⋐ Σ and we choose a radius $0<r_{2}^{\prime }<r_{2}$, a sector Σ^″ such that Σ^′ ⋐ Σ^″ ⋐ Σ and a positive constant δ > 0 small enough so that, for all t ∈Σ^′, the closed disc centered at t with radius |t|δ be contained in Σ^″. Then, the Cauchy integral formula implies

$$\begin{array}{@{}rcl@{}} {\partial_{t}^{J}}u(t,x)&=&\frac{J!}{(2i\pi)^{2}}\underset{\left\vert x^{\prime}-x\right\vert=r_{2}-r_{2}^{\prime}}{{\int}_{\left\vert t^{\prime}-t\right\vert=\left\vert t\right\vert\delta}}\frac{u(t^{\prime},x^{\prime})}{(t^{\prime}-t)^{J + 1}(x^{\prime}-x)}dt^{\prime}dx^{\prime}\\ &=&\frac{J!}{(2i\pi)^{2}}\underset{\left\vert x^{\prime}-x\right\vert=r_{2}-r_{2}^{\prime}}{{\int}_{\left\vert t^{\prime}-t\right\vert=\left\vert t\right\vert\delta}}\left( u(t^{\prime},x^{\prime})-{\sum}_{j = 0}^{J-1}u_{j,\ast}(x^{\prime})\frac{t^{\prime j}}{j!}\right)\frac{dt^{\prime}dx^{\prime}}{(t^{\prime}-t)^{J + 1}(x^{\prime}-x)} \end{array} $$

for all $J\geqslant 0$, all t ∈Σ^′ and all $\left \vert x\right \vert \leqslant r_{2}^{\prime }$. Indeed, the sum is 0 when J = 0 and the J-th derivative of a polynomial of degree J − 1 is 0 too when $J\geqslant 1$. Hence,

$$\begin{array}{@{}rcl@{}} \left\vert{\partial_{t}^{J}}u(t,x)\right\vert&\leqslant& CK^{J}{\Gamma}(1+sJ \null)J!\frac{\left\vert t\right\vert^{J}(1+\delta)^{J}}{\left\vert t\right\vert^{J}\delta^{J}}\\ &\leqslant& C^{\prime}K^{{\prime}J}{\Gamma}(1+(s + 1)J \null) \end{array} $$

with C^′ = eC and $K^{\prime }=e^{s + 1}K\left (1+\frac {1}{\delta }\right )$. Indeed, we have previously seen in the proof of Proposition 7 that Γ(1 + sJ)J! ≤ e^1+(s+ 1)JΓ(1 + (s + 1)J). This proves Condition 2 and, consequently, the necessary condition.⊲ Sufficient condition. Let us now suppose that Conditions 1 and 2 are satisfied and let us prove condition (1) of Definition 5. To do that, let us consider a proper subsector Σ^′ ⋐ Σ.

For any fixed |x| ≤ r₂, the map t↦u(t, x) admits the Taylor expansion with integral remainder

$$ u(t,x)-\sum\limits_{j = 0}^{J-1}\frac{\partial^{j} u}{\partial t^{j}}(t_{0},x)\frac{(t-t_{0})^{j}}{j!}={\int}_{t_{0}}^{t}\frac{(t-t^{\prime})^{J-1}}{(J-1)!}\frac{\partial^{J} u}{\partial t^{J}}(t^{\prime},x)dt^{\prime} $$

(3)

for all $J\geqslant 1$, all t ∈Σ^′ and all t₀ ∈Σ^′. Due to Condition 1, $\underset {t_{0}\in {\Sigma }^{\prime }}{\underset {t_{0}\rightarrow 0}{\lim }} \frac {\partial ^{j} u}{\partial t^{j}}(t_{0},x)$ exists for all $j\geqslant 0$ and is equal to u_j,∗(x). Therefore, the limits of the left-hand and of the right-hand sides of (3) both exist when t₀ → 0 and we have

$$u(t,x)-{\sum}_{j = 0}^{J-1}u_{j,\ast}(x)\frac{t^{j}}{j!}={{\int}_{0}^{t}}\frac{(t-t^{\prime})^{J-1}}{(J-1)!}\frac{\partial^{J} u}{\partial t^{J}}(t^{\prime},x)dt^{\prime} $$

for all $J\geqslant 1$, all t ∈Σ^′ and all |x| ≤ r₂. Hence, applying Condition 2:

$$\sup_{\left\vert x\right\vert\leqslant r_{2}}\left\vert u(t,x)-\sum\limits_{j = 0}^{J-1}u_{j,\ast}(x)\frac{t^{j}}{j!}\right\vert\leqslant \underset{\left\vert x\right\vert\leqslant r_{2}}{\underset{t^{\prime}\in{\Sigma}^{\prime}}{\sup}} \left\vert\frac{\partial^{J} u}{\partial t^{J}}(t^{\prime},x)\right\vert\frac{\left\vert t\right\vert^{J}}{J!}\leqslant CK^{J}\frac{{\Gamma}(1+(s + 1)J \null)}{J!}\left\vert t\right\vert^{J} $$

for all $J\geqslant 1$ and all t ∈Σ^′. Condition (3) follows then from the inequality

$$\frac{{\Gamma}(1+(s + 1)J \null)}{J!}\leqslant 2^{(S + 1)J}{\Gamma}(1+sJ \null)\quad,S\in\mathbb N,S\geqslant s $$

which stems from the relations

$${\Gamma}(1+(s + 1)J \null)={\Gamma}(1+sj \null){\prod}_{j = 1}^{J}(sJ+j)\leqslant{\Gamma}(1+sJ \null){\prod}_{j = 1}^{J}(SJ+j) $$

and

$$\frac{\displaystyle{\prod}_{j = 1}^{J}(SJ+j)}{J!}=\left( \begin{array}{c}(S + 1)J\\J \end{array}\right)\leqslant\sum\limits_{k = 0}^{(S + 1)J}\left( \begin{array}{c}(S + 1)J\\k \end{array}\right)= 2^{(S + 1)J}. $$

This proves the sufficient condition; hence, Proposition 8. □

In the sequel, we denote by

• $\overline {\mathcal {A}}_{s}({\Sigma }, D_{\rho _{2}})$ the set of all the functions which are s-Gevrey asymptotic on Σ to a formal series of $\mathcal {O}(D_{\rho _{2}})[[t]]$;

• $T_{s;{\Sigma },D_{\rho _{2}}}:\overline {\mathcal {A}}_{s}({\Sigma }, D_{\rho _{2}})\longrightarrow \mathcal {O}(D_{\rho _{2}})[[t]]_{s}$ the map which assigns to each $u(t,x)\in \overline {\mathcal {A}}_{s}({\Sigma }, D_{\rho _{2}})$ its s-Gevrey asymptotic series.

Observe that $T_{s;{\Sigma },D_{\rho _{2}}}$ is well-defined due to Remark 12 and Proposition 8. Following Proposition 9 specifies the algebraic properties of $\overline {\mathcal {A}}_{s}({\Sigma }, D_{\rho _{2}})$ and $T_{s;{\Sigma },D_{\rho _{2}}}$.

Proposition 9

Let$s\geqslant 0$andΣ be an open sectorwith vertex$0\in \mathbb {C}$.

• 1. $(\overline {\mathcal {A}}_{s}({\Sigma }, D_{\rho _{2}}),\partial _{t},\partial _{x})$isa$\mathbb {C}$-differentialalgebra stable under the anti-derivations$\partial _{t}^{-1}$and$\partial _{x}^{-1}$.

• 2. The map$T_{s;{\Sigma },D_{\rho _{2}}}:\overline {\mathcal {A}}_{s}({\Sigma }, D_{\rho _{2}})\longrightarrow \mathcal {O}(D_{\rho _{2}})[[t]]_{s}$isa homomorphism of$\mathbb {C}$-differentialalgebras for the derivations∂_tand∂_x.Moreover, it commutes with the anti-derivations$\partial _{t}^{-1}$and$\partial _{x}^{-1}$.

Proof

The proof is the same that the one given in [43, Prop. 2]. □

1.2 The s-Gevrey Borel-Ritt Theorem

Theorem 4

Supposing that Σ has opening ≤ πs.Then, the map$T_{s;{\Sigma },D_{\rho _{2}}}$isonto.

Proof

It is sufficient to consider a sector Σ with opening πs. Moreover, by means of a rotation, we can besides assume that Σ is bisected by the direction 𝜃 = 0. We denote by R its radius.

• Let $\widetilde {u}(t,x)=\displaystyle {\sum }_{j\geqslant 0}u_{j,\ast }(x)\frac {t^{j}}{j!}\in \mathcal {O}(D_{\rho _{2}})[[t]]_{s}$ a s-Gevrey formal series. By assumption, the coefficients u_j,∗(x) satisfy the following two conditions:

  • $u_{j,\ast }(x)\in \mathcal {O}(D_{\rho _{2}})$ for all $j\geqslant 0$,

  • There exist 0 < r₂ < ρ₂, C > 0 and K > 0 such that |u_j,∗(x)| ≤ CK^jΓ(1 + (s + 1)j) for all $j\geqslant 0$ and |x|≤ r₂.

  Therefore, the series $\widehat {u}(\tau ,x)=\displaystyle {\sum }_{j\geqslant 0}\frac {u_{j,\ast }(x)\tau ^{j}}{{\Gamma }(1+sj)j!}$ converges for all $(\tau ,x)\in D_{\rho }\times D_{r_{2}}$, where ρ is the radius of convergence of $\displaystyle {\sum }_{j\geqslant 0}\frac {{\Gamma }(1+(s + 1)j)}{{\Gamma }(1+sj)j!}(K\tau )^{j}$.

• Let us now fix b ∈ D_ρ, b > 0, and let us consider the holomorphic function $u(t,x)\in \mathcal {O}({\Sigma }\times D_{r_{2}})$ defined by

  $$u(t,x)=t^{-k}{\int}_{0}^{b^{k}}\widehat{u}(\xi^{s},x)e^{-\xi/t^{k}}d\xi\text{, where }s=\frac{1}{k}\text{ and }\xi=\tau^{k}. $$

  We shall prove below that u(t, x) is s-Gevrey asymptotic to $\widetilde {u}(t,x)$ on Σ.

• Let $0<r_{2}^{\prime }<r_{2}$. For any $0<\delta <\frac {\pi }{2}$ and 0 < R^′ < R, we denote by Σ_δ the proper subsector of Σ defined by

  $${\Sigma}_{\delta}=\left\{t\in\mathbb{C};\left\vert\arg(t)\right\vert<\frac{\pi}{2k}-\frac{\delta}{k}\text{ and }0<|t|<R^{\prime}\right\}. $$

Let $J\geqslant 1$ and $(t,x)\in {\Sigma }_{\delta }\times \overline {D}_{r_{2}^{\prime }}$ be. From the relation

$$t^{j}=t^{-k}{\int}_{0}^{+\infty}\frac{\xi^{sj}}{{\Gamma}(1+sj)}e^{-\xi/t^{k}}d\xi\text{, }j\geqslant 0 $$

(see [2, pp. 78–79] for instance), we first have

$$\begin{array}{@{}rcl@{}} u(t,x)-\sum\limits_{j = 0}^{J-1}u_{j,\ast}(x)\frac{t^{j}}{j!}&=&t^{-k}{\int}_{0}^{b^{k}}\left( \sum\limits_{j\geqslant 0}\frac{u_{j,\ast}(x)}{{\Gamma}(1+sj)j!}\xi^{sj}e^{-\xi/t^{k}}\right)d\xi\\ &&-\sum\limits_{j = 0}^{J-1}\frac{u_{j,\ast}(x)}{j!}t^{-k}{\int}_{0}^{+\infty}\frac{\xi^{sj}}{{\Gamma}(1+sj)}e^{-\xi/t^{k}}d\xi. \end{array} $$

Since

$$t\in{\Sigma}_{\delta}\Rightarrow|\arg(t)|<\frac{\pi}{2}\Rightarrow\Re(t)>0\Rightarrow\left\vert\xi^{sj}e^{-\xi/t^{k}}\right\vert=|\xi|^{sj}e^{-\xi\frac{\Re(t^{k})}{|t|^{2k}}}\leqslant b^{j} $$

for all ξ ∈ [0, b^k], the series $\displaystyle {\sum }_{j\geqslant 0}\frac {u_{j,\ast }(x)}{{\Gamma }(1+sj)j!}\xi ^{sj}e^{-\xi /t^{k}}$ converges normally on [0, b^k]. Therefore, we can permute the sum and the integral. Hence,

$$\begin{array}{@{}rcl@{}} u(t,x)-\sum\limits_{j = 0}^{J-1}u_{j,\ast}(x)\frac{t^{j}}{j!}&=&\sum\limits_{j\geqslant J}\frac{u_{j,\ast}(x)}{{\Gamma}(1+sj)j!}t^{-k}{\int}_{0}^{b^{k}}\xi^{sj}e^{-\xi/t^{k}}d\xi\\ &&-\sum\limits_{j = 0}^{J-1}\frac{u_{j,\ast}(x)}{{\Gamma}(1+sj)j!}t^{-k}{\int}_{b^{k}}^{+\infty}\xi^{sj}e^{-\xi/t^{k}}d\xi. \end{array} $$

Let us now observe that the inequalities (ξ/b^k)^sj ≤ (ξ/b^k)^Js hold both when ξ ≤ b^k and $j\geqslant J$ and when $\xi \geqslant b^{k}$ and j < J. This brings us then to the following:

$$\begin{array}{@{}rcl@{}} \left\vert u(t,x)-\sum\limits_{j = 0}^{J-1}u_{j,\ast}(x)\frac{t^{j}}{j!}\right\vert&\leqslant&\sum\limits_{j\geqslant J}\frac{b^{j-J}\left\vert u_{j,\ast}(x)\right\vert}{{\Gamma}(1+sj)j!}|t|^{-k}{\int}_{0}^{b^{k}}\xi^{sJ}e^{-\xi\Re(1/t^{k})}d\xi\\ &&+\sum\limits_{j = 0}^{J-1}\frac{ b^{j-J}\left\vert u_{j,\ast}(x)\right\vert}{{\Gamma}(1+sj)j!}|t|^{-k}{\int}_{b^{k}}^{+\infty}\xi^{sJ}e^{-\xi\Re(1/t^{k})}d\xi\\ &=&\sum\limits_{j\geqslant 0}\frac{ b^{j-J}\left\vert u_{j,\ast}(x)\right\vert}{{\Gamma}(1+sj)j!}|t|^{-k}{\int}_{0}^{+\infty}\xi^{sJ}e^{-\xi\Re(1/t^{k})}d\xi\\ &\leqslant&\sum\limits_{j\geqslant 0}\frac{ b^{j-J}\left\vert u_{j,\ast}(x)\right\vert}{{\Gamma}(1+sj)j!}|t|^{-k}{\int}_{0}^{+\infty}\xi^{sJ}e^{-\xi\sin(\delta)/|t|^{k})}d\xi. \end{array} $$

Observe that the last inequality stems from the fact that t ∈Σ_δ implies

$$\Re\left( \frac{1}{t^{k}}\right)=\frac{\cos\left( \arg(t^{k})\right)}{|t|^{k}}\geqslant \frac{\cos\left( \frac{\pi}{2}-\delta\right)}{|t|^{k}}=\frac{\sin(\delta)}{|t|^{k}}. $$

Setting then $u=\frac {\xi \sin (\delta )}{|t|^{k}}$, we obtain

$$\begin{array}{@{}rcl@{}} \left\vert u(t,x)-\sum\limits_{j = 0}^{J-1}u_{j,\ast}(x)\frac{t^{j}}{j!}\right\vert &\leqslant&\sum\limits_{j\geqslant 0}\frac{ b^{j-J}\left\vert u_{j,\ast}(x)\right\vert|t|^{J}}{{\Gamma}(1+sj)j!(\sin(\delta))^{sJ+ 1}}{\int}_{0}^{+\infty}u^{sJ}e^{-u}du\\ &=&\sum\limits_{j\geqslant 0}\frac{ b^{j-J}\left\vert u_{j,\ast}(x)\right\vert}{{\Gamma}(1+sj)j!(\sin(\delta))^{sJ+ 1}}{\Gamma}(1+sJ)|t|^{J}, \end{array} $$

where, according to the choice of b (see the beginning of the proof), we have

$$\displaystyle{\sum}_{j\geqslant 0}\frac{ b^{j}\left\vert u_{j,\ast}(x)\right\vert}{{\Gamma}(1+sj)j!}\leqslant C\displaystyle{\sum}_{j\geqslant 0}\frac{{\Gamma}(1+(s + 1)j)}{{\Gamma}(1+sj)j!}(Kb)^{j}<+\infty.$$

Consequently, we finally get

$$\left\vert u(t,x)-\sum\limits_{j = 0}^{J-1}u_{j,\ast}(x)\frac{t^{j}}{j!}\right\vert\leqslant C^{\prime}K^{{\prime}J}{\Gamma}(1+sJ)|t|^{J}, $$

with $C^{\prime }=\frac {C}{\sin (\delta )}\displaystyle \sum \limits _{j\geqslant 0}\frac {{\Gamma }(1+(s + 1)j)}{{\Gamma }(1+sj)j!}(Kb)^{j}$ and $K^{\prime }=\frac {1}{b(\sin (\delta ))^{s}}$. The constants C^′ and K^′ depend on Σ_δ and on the choice of b, but are independent of t and x. This achieves the proof. □