Trotter product formula and linear evolution equations on Hilbert spaces On the occasion of the 100th birthday of Tosio Kato

TL;DR

This paper investigates the approximation of solutions to linear evolution equations on Hilbert spaces using Trotter product formulas, establishing convergence conditions and rates based on operator regularity and Hölder continuity.

Contribution

It provides new conditions under which the Trotter product formula converges for evolution equations with time-dependent operators on Hilbert spaces, including explicit convergence rates.

Findings

Convergence of the approximation when Hölder exponent exceeds 2α - 1

Operator norm approximation of solution operators by semigroup products

Explicit convergence rate tied to Hölder continuity

Abstract

The paper is devoted to evolution equations of the form $\partial$ $\partial$t u(t) = --(A + B(t))u(t), t $\in$ I = [0, T ], on separable Hilbert spaces where A is a non-negative self-adjoint operator and B($\times$) is family of non-negative self-adjoint operators such that dom(A $\alpha$) $\subseteq$ dom(B(t)) for some $\alpha$ $\in$ [0, 1) and the map A --$\alpha$ B($\times$)A --$\alpha$ is H{\"o}lder continuous with the H{\"o}lder exponent $\beta$ $\in$ (0, 1). It is shown that the solution operator U(t, s) of the evolution equation can be approximated in the operator norm by a combination of semigroups generated by A and B(t) provided the condition $\beta$ > 2$\alpha$ -- 1 is satisfied. The convergence rate for the approximation is given by the H{\"o}lder exponent $\beta$. The result is proved using the evolution semigroup approach.

Equations

$$\frac{\partial }{\partial t}u(t)=-(A+B(t))u(t),t\in \mathcal{I}=[0,T],$$

$$\begin{array}{rl}\frac{\partial u(t)}{\partial t}=& -C(t)u(t),u(s)=x_s,s\in [0,T),\\ C(t)=& A+B(t),\end{array}t\in \mathcal{I}:=[0,T],$$

$$C_{\alpha }:=\underset{t\in \mathcal{I}}{\mathrm{esssup}}\Vert B(\cdot )A^{-\alpha }\Vert <\infty .$$

$$\Vert A^{-\alpha }(B(t)-B(s))A^{-\alpha }\Vert \le L_{\alpha ,\beta }|t-s{|}^{\beta },(t,s)\in \mathcal{I}\times \mathcal{I},$$

$$\begin{array}{rl}U(t,t)=& I,\text{mbox}fort\in \mathcal{I},\\ U(t,r)U(r,s)=& U(t,s),\text{mbox}fort,r,s\in \mathcal{I}\text{mbox}withs\le r\le t,\end{array}$$

$$s=:t_0<t_1<\dots <t_{n-1}<t_n:=t,t_j:=s+j\frac{t-s}{n},$$

$$\begin{array}{rl}{G}_j(t,s;n):=& e^{-\frac{t-s}{n}A}{e}^{-\frac{t-s}{n}B(t_j)},j=0,1,2,\dots ,n,\\ V_n(t,s):=& G_{n-1}(t,s;n)G_{n-1}(t,s;n)\times \cdots \times G_2(t,s;n)G_0(t,s;n),\end{array}$$

$$\underset{(t,s)\in \Delta }{\mathrm{esssup}}\Vert V_n(t,s)-U(t,s)\Vert \le \frac{R_{\beta }}{n^{\beta }},n\in \mathbb{N},$$

$$\underset{(t,s)\in \Delta }{\mathrm{esssup}}\Vert V_n(t,s)-U(t,s)\Vert \le \frac{R_{\gamma }}{n^{\gamma }},n\in \mathbb{N}.$$

$$\underset{t\in \mathcal{I}}{\mathrm{esssup}}\Vert V_n(t,0)-U(t,0)\Vert \le {{\rm Y}}_0\frac{\log (n)}{n},n=2,3,\dots .$$

$$\underset{t\in \mathcal{I}}{\mathrm{esssup}}\Vert V_n(t,0)-U(t,0)\Vert \le \frac{R_{\gamma }}{n^{\gamma }},n\in \mathbb{N}.$$

$$({\mathcal{U}}_0(\tau )f)(t):=e^{-\tau A}{\chi }_{\mathcal{I}}(t-\tau )f(t-\tau ),f\in L^2(\mathcal{I},\mathfrak{H}),$$

$$\mathcal{K}={\mathcal{K}}_0+\mathcal{B},\mathrm{dom}(\mathcal{K})=\mathrm{dom}({\mathcal{K}}_0)\cap \mathrm{dom}(\mathcal{B}),$$

$$\text{mbox}\mathrm{s}-\underset{n\to \infty }{\lim }{\left(e^{-\frac{\tau }{n}{\mathcal{K}}_0}{e}^{-\frac{\tau }{n}\mathcal{B}}\right)}^n=e^{-\tau \mathcal{K}},\tau \in {\mathbb{R}}_{+}:=[0,\infty ),$$

$$\underset{\tau \in {\mathbb{R}}_{+}}{\sup }\Vert {\left(e^{-\frac{\tau }{n}{\mathcal{K}}_0}{e}^{-\frac{\tau }{n}\mathcal{B}}\right)}^n-e^{-\tau \mathcal{K}}\Vert \le \frac{R_{\beta }}{n^{\beta }},n\in \mathbb{N},$$

$$(\mathcal{U}(\tau )f)(t)=U(t,t-\tau ){\chi }_{\mathcal{I}}(t-\tau )f(t-\tau ),f\in L^2(\mathcal{I},\mathfrak{H}).$$

$$\underset{(t,s)\in {\Delta }_0}{\mathrm{esssup}}\Vert V_n(t,s)-U(t,s)\Vert \le \frac{R_{\beta }}{n^{\beta }},n\in \mathbb{N},$$

$$(e^{-\tau D_0}f)(t)={\chi }_{\mathcal{I}}(t-\tau )f(t-\tau ),t\in \mathcal{I},\tau \in {\mathbb{R}}_{+},f\in L^2(\mathcal{I},\mathfrak{H}).$$

$$\begin{array}{rl}(D_{0}f)(t):=& \frac{\partial }{\partial t}f(t),\\ f\in \mathrm{dom}(D_0):=& W_0^{1,2}(\mathcal{I},\mathfrak{H})=\{f\in W^{1,2}(\mathcal{I},\mathfrak{H}):f(0)=0\}.\end{array}$$

$$\begin{split}({\mathcal{A}}f)(t):=&Af(t),\\ f\in\mathrm{dom}({\mathcal{A}}):=&\left\{f\in L^{2}({\mathcal{I}},{\mathfrak{H}}):\begin{matrix}&f(t)\in\mathrm{dom}(A)\quad\mbox{for a.e. $t\in{\mathcal{I}}$}\\ &Af(t)\in L^{2}({\mathcal{I}},{\mathfrak{H}}).\end{matrix}\right\}\end{split}$$

$$((\mathcal{A}-z{)}^{-1}f)(t)=(A-z{)}^{-1}f(t),t\in {\mathcal{I}}_0,f\in L^2(\mathcal{I},\mathfrak{H}),z\in \rho (\mathcal{A})=\rho (A),$$

$$(e^{-\tau \mathcal{A}}f)(t)=e^{-\tau A}f(t),t\in \mathcal{I},f\in L^2(\mathcal{I},\mathfrak{H}),\tau \in {\mathbb{R}}_{+}.$$

$${\mathcal{U}}_0(\tau ):=e^{-\tau D_0}{e}^{-\tau \mathcal{A}},\tau \in {\mathbb{R}}_{+}.$$

$${\mathcal{K}}_0:=D_0+\mathcal{A},\mathrm{dom}({\mathcal{K}}_0):=\mathrm{dom}(D_0)\cap \mathrm{dom}(\mathcal{A}).$$

$$\begin{split}({\mathcal{B}}f)(t):=&B(t)f(t)\\ f\in\mathrm{dom}({\mathcal{B}}):=&\left\{f\in L^{2}({\mathcal{I}},{\mathfrak{H}}):\begin{matrix}&f(t)\in\mathrm{dom}(B(t))\quad\mbox{for a.e. $t\in{\mathcal{I}}$}\\ &B(t)f(t)\in L^{2}({\mathcal{I}},{\mathfrak{H}})\end{matrix}\right\}\;.\end{split}$$

$$((\mathcal{B}-z{)}^{-1}f)(t)=(B(t)-z{)}^{-1}f(t),f\in L^2(\mathcal{I},\mathfrak{H}),z\in {\mathbb{C}}_{\pm },$$

$$(e^{-\tau \mathcal{B}}f)(t)=e^{-\tau B(t)}f(t),f\in L^2(\mathcal{I},\mathfrak{H}),$$

$$\mathcal{K}:={\mathcal{K}}_0+\mathcal{B},\mathrm{dom}(\mathcal{K}):=\mathrm{dom}({\mathcal{K}}_0)\cap \mathrm{dom}(\mathcal{B}),$$

$$B(t)A^{-{\alpha }^{\prime }}=B(t)A^{-\alpha }{A}^{-({\alpha }^{\prime }-\alpha )}$$

$$C_{{\alpha }^{\prime }}:=\underset{t\in \mathcal{I}}{\mathrm{esssup}}\Vert B(t)A^{-{\alpha }^{\prime }}\Vert \le \underset{t\in \mathcal{I}}{\mathrm{esssup}}\Vert B(t)A^{-\alpha }\Vert =C_{\alpha }<\infty .$$

Full text

[TABLE]

on separable Hilbert spaces where $A$ is a non-negative self-adjoint operator and $B(\cdot)$ is family of non-negative self-adjoint operators such that $\mathrm{dom}(A^{\alpha})\subseteq\mathrm{dom}(B(t))$ for some ${\alpha}\in[0,1)$ and the map $A^{-{\alpha}}B(\cdot)A^{-{\alpha}}$ is Hölder continuous with the Hölder exponent ${\beta}\in(0,1)$. It is shown that the solution operator $U(t,s)$ of the evolution equation can be approximated in the operator norm by a combination of semigroups generated by $A$ and $B(t)$ provided the condition ${\beta}>2{\alpha}-1$ is satisfied. The convergence rate for the approximation is given by the Hölder exponent ${\beta}$. The result is proved using the evolution semigroup approach.

1 Introduction

A closer look to Kato’s work shows that abstract evolution equations and Trotter product formula were topics of high interest for Kato. Already at the beginning of his scientific career Kato was interested in evolution equations Kato1953 ; Kato1956 . This interest has lasted a lifetime Kato1961 ; Kato1965 ; Kato1970 ; Kato1973 ; Kato2011a ; Kato2011b ; KatoTan1962 . Another topic of great interest for him was the so-called Trotter product formula Kato1978 ; Kato1978b ; Kato1974 ; Kato1978c . Even the paper Kato1978 has inspired further developments in this field ITTZ2001 .

The topic of the present paper is to link evolution equations with the Trotter product formula. To this end we consider an abstract evolution equation of type

[TABLE]

on the separable Hilbert space ${\mathfrak{H}}$. Evolution equations of that type on Hilbert or Banach spaces are widely investigated, cf. AcquistapaceTerreni1984 ; AcquistapaceTerreni1985 ; Amann1988 ; Amann1987 ; ArendtDierOuhabaz2014 ; KatoTan1962 ; Lunardi1987 ; MonniauxRhandi2000 ; Tan1959 ; Tan1960a ; Tan1960b ; Tan1961 ; Tan1967 ; Tan1967b ; Yagi1976 ; Yagi1977 ; Yagi1988 ; Yagi1989 ; Yagi1990 or the books Amann1995 ; Tan1979 ; Yagi2010 . We consider the equation (1.1) under the following assumptions.

Assumption 1.1

2. (S1)

The operator $A$ is self-adjoint in the Hilbert space ${\mathfrak{H}}$ such that $A\geq I$. Let $\{B(t)\}_{t\in{\mathcal{I}}}$ be a family of non-negative self-adjoint operators in ${\mathfrak{H}}$ such that the function $(I+B(\cdot))^{-1}:{\mathcal{I}}\longrightarrow{\mathcal{L}}({\mathfrak{H}})$ is strongly measurable. 3. (S2)

There is an ${\alpha}\in[0,1)$ such that for a.e. $t\in{\mathcal{I}}$ the inclusion $\mathrm{dom}(A^{\alpha})\subseteq\mathrm{dom}(B(t))$ holds. Moreover, the function $B(\cdot)A^{-{\alpha}}:{\mathcal{I}}\longrightarrow{\mathcal{L}}({\mathfrak{H}})$ is strongly measurable and essentially bounded, i.e.

[TABLE] 4. (S3)

The map $A^{-{\alpha}}B(\cdot)A^{-{\alpha}}:{\mathcal{I}}\longrightarrow{\mathcal{L}}({\mathfrak{H}})$ is Hölder continuous, i.e, for some ${\beta}\in(0,1)$ there is a constant $L_{{\alpha},{\beta}}>0$ such that the estimate

[TABLE]

holds. $\triangle$

Notice that under the assumption (S2) the operator $C(t)$ is also an invertible non-negative self-adjoint operator for each $t\in{\mathcal{I}}$. Assumptions of that type were made in FujTan1973 ; IchinoseTamura1998 ; NeiStephZagr2016 ; NeiStephZagr2017 ; Yagi1990 . One checks that the assumptions (S1)-(S3) and the additional assumption ${\beta}>{\alpha}$ imply the assumptions (I), (VI) and (VII) of Yagi1990 for the family $\{C(t)\}_{\in{\mathcal{I}}}$. Hence, Proposition 3.1 and Theorem 3.2 of Yagi1990 yield the existence of a so-called solution (or evolution) operator for the evolution equation (1.1), i.e., a strongly continuous, uniformly bounded family of bounded operators $\{U(t,s)\}_{(t,s)\in{\Delta}}$, ${\Delta}~{}:=~{}\{(t,s)\in{\mathcal{I}}\times{\mathcal{I}}:0\leq s\leq t\leq T\}$, such that the conditions

[TABLE]

are satisfied and $u(t)=U(t,0)x$ is for every $x\in{\mathfrak{H}}$ a strict solution of (1.1), see Definition 1.1 of Yagi1990 . Because the involved operators are self-adjoint and non-negative one checks that the solution operator consists of contractions.

The aim of the present paper is to analyze the convergence of the following approximation to the solution operator $\{U(t,s)\}_{(t,s)\in{\Delta}}$. Let

[TABLE]

$j=\{0,1,2,\ldots,n\}$, $n\in{\mathbb{N}}$, be a partition of the interval $[s,t]$. Let

[TABLE]

$n\in{\mathbb{N}}$. The main result in the paper is the following. If the assumptions (S1)-(S3) are satisfied and in addition the condition ${\beta}>{\alpha}$ holds, then the solution operator $\{U(t,s)\}_{(t,s)\in{\Delta}}$ of Yagi1990 admits the approximation

[TABLE]

with some constant $R_{\beta}>0$. The result shows that the convergence of the approximation $\{V_{n}(t,s)\}_{(t,s)\in{\Delta}}$ is determined by the smoothness of the perturbation $B(\cdot)$.

If the map $A^{-{\alpha}}B(\cdot)A^{-{\alpha}}:{\mathcal{I}}\longrightarrow{\mathcal{L}}({\mathfrak{H}})$ is Lipschitz continuous, then the map is of course Hölder continuous with any exponent ${\gamma}\in({\alpha},1)$. Hence from (1.7) it immediately follows that for any ${\gamma}\in({\alpha},1)$ there is a constant $R_{\gamma}$ such that

[TABLE]

In particular, for any ${\gamma}$ close to one the estimate (1.8) holds.

In IchinoseTamura1998 the Lipschitz case was considered. It was shown that there is a constant $\Upsilon_{0}>0$ such that the estimate

[TABLE]

holds. It is obvious that the estimate (1.9) is stronger than

[TABLE]

(which follows from (1.8)) for any ${\gamma}$ independent of how close it is to one.

To prove (1.7) we use the so-called evolution semigroup approach which allows not only to verify the estimate (1.7) but also to generalise it. The approach is quite different from the technique used in IchinoseTamura1998 ; Yagi1990 . We have successfully applied this approach already in NeiStephZagr2016 and NeiStephZagr2017 . The key idea is to forget about the evolution equation (1.1) and to consider instead of it the operators ${\mathcal{K}}_{0}$ and ${\mathcal{K}}$ on ${\mathfrak{K}}=L^{2}({\mathcal{I}},{\mathfrak{H}})$. The operator ${\mathcal{K}}_{0}$ is the generator of the contraction semigroup $\{{\mathcal{U}}_{0}({\tau})\}_{{\tau}\in{\mathbb{R}}_{+}}$,

[TABLE]

and ${\mathcal{K}}$ is given by

[TABLE]

where ${\mathcal{B}}$ is the multiplication operator induced by the family $\{B(t)\}_{\in{\mathcal{I}}}$ in $L^{2}({\mathcal{I}},{\mathfrak{H}})$ which is self-adjoint and non-negative, for more details see Section 2. It turns out that under the assumptions (S1) and (S2) the operator ${\mathcal{K}}$ is the generator of a contraction semigroup $\{{\mathcal{U}}({\tau})\}_{{\tau}\in{\mathbb{R}}_{+}}$ on $L^{2}({\mathcal{I}},{\mathfrak{H}})$. For the pair $\{{\mathcal{K}}_{0},{\mathcal{B}}\}$ we consider the Lie-Trotter product formula. From the original paper of Trotter Trotter1959 one gets that

[TABLE]

holds uniformly in ${\tau}$ on any bounded interval of ${\mathbb{R}}_{+}$. Since $e^{-{\tau}{\mathcal{K}}_{0}}=0$ and $e^{-{\tau}{\mathcal{K}}}=0$ for ${\tau}\geq T$ one gets even uniformly in ${\tau}\in{\mathbb{R}}_{+}$.

Previously it was shown that under certain assumptions the strong convergence can be improved to operator-norm convergence on Hilbert spaces, see CachNeiZag2001 ; CachNeiZag2002 ; ITTZ2001 ; NeidhardtZagrebnov1998 ; Rogava1993 as well as on Banach spaces, see CachZag2001 . For an overview the reader is referred to NeiStephZagr2018 . To consider the Trotter product formula for evolution equations is relatively new and was firstly realized in NeiStephZagr2016 ; NeiStephZagr2017 for Banach spaces.

In the following we improve the convergence (1.11) to operator-norm convergence. We show that under the assumptions (S1)-(S3) and ${\beta}>2{\alpha}-1$ there is a constant $R_{\beta}>0$ such that

[TABLE]

holds.

It turns out that ${\mathcal{K}}$ is the generators of an evolution semigroup. This means, there is a propagator $\{U(t,s)\}_{(t,s)\in{\Delta}_{0}}$, ${\Delta}_{0}:=\{(t,s)\in{\mathcal{I}}_{0}\times{\mathcal{I}}_{0}:s\leq t\}$, ${\mathcal{I}}_{0}=(0,T]$, such that the contraction semigroup $\{{\mathcal{U}}({\tau})=e^{-{\tau}{\mathcal{K}}}\}_{{\tau}\in{\mathbb{R}}_{+}}$ admits the representation

[TABLE]

We recall that a strongly continuous, uniformly bounded family of bounded operators $\{U(t,s)\}_{(t,s)\in{\Delta}_{0}}$ is called a propagator if (1.4) is satisfied for ${\mathcal{I}}_{0}$ and ${\Delta}_{0}$ instead of ${\mathcal{I}}$ and ${\Delta}$, respectively. Roughly speaking, a propagator is a solution operator restricted to ${\Delta}_{0}$ where the assumption that $U(t,0)x$ should be a strict solution is dropped. Obviously, the notion of a propagator is weaker then that one of a solution operator. For its existence one needs only the assumptions (S1) and (S2), see Theorem 4.4 and 4.5 in NeiStephZagr2016 or Theorem 3.3 NeiStephZagr2017 . Of course, the propagator coincides with the solution operator of Yagi1990 if the assumptions (S1)-(S3) are satisfied and ${\beta}>{\alpha}$.

By Proposition 3.8 of NeiStephZagr2018 and (1.12) we immediately get that under the assumptions (S1)-(S3) and ${\beta}>2{\alpha}-1$ the estimate

[TABLE]

holds, where the constant $R_{\beta}$ is that one of (1.12). Notice that the condition ${\beta}>2{\alpha}-1$ is weaker than ${\beta}>{\alpha}$, i.e., if ${\beta}>{\alpha}$, then ${\beta}>2{\alpha}-1$ holds. If ${\alpha}$ satisfies the condition $\tfrac{1+{\beta}}{2}>{\alpha}>{\beta}$, then the assumptions (I), (VI) and (VII) of Yagi1990 for the family $\{C(t)\}_{\in{\mathcal{I}}}$ are not valid but nevertheless we get an approximation of the corresponding propagator $\{U(t,s)\}_{(t,s)\in{\Delta}_{0}}$.

The results are stronger than those in NeiStephZagr2016 ; NeiStephZagr2017 for Banach spaces. In NeiStephZagr2016 a convergence rate $O(n^{-({\beta}-{\alpha})})$ was found, whereas in NeiStephZagr2017 the Lipschitz case has been considered and the rate $O(n^{-(1-{\alpha})})$ for ${\alpha}\in(\tfrac{1}{2},1)$ was proved.

It turns out that the result (1.7) can be hardly improved. Indeed in NeiStephZagr2018b the simple case ${\mathfrak{H}}:={\mathbb{C}}$ and $A=1$ was considered. In that case the family $\{B(t)\}_{t\in{\mathcal{I}}}$ reduces to a non-negative bounded measurable function: $b(\cdot):{\mathcal{I}}\longrightarrow{\mathbb{R}}$ which has to be Hölder continuous with exponent ${\beta}\in(0,1)$. For that case it was found in NeiStephZagr2018b that the convergence rate is $O(n^{-{\beta}})$ which coincides with (1.7). For the Lipschitz case it was found $O(n^{-1})$ which suggests that (1.8) and (1.9) might be not optimal.

The paper is organised as follows. In Section 2 we give a short introduction into evolution semigroups. For more details the reader is referred to Nei1981 ; NeiStephZagr2016 ; NeiZag2009 ; Nickel1996 . The results are proven in Section 3. In Section 3.1 we prove auxiliary results which are necessary to prove the main results of Section 3.2.

Notation: Spaces, in particular, Hilbert are denoted by Gothic capital letters like ${\mathfrak{H}}$, ${\mathfrak{K}}$, etc. Operators are denoted by Latin or italic capital letters. The Banach space of bounded operators on space is denoted by ${\mathcal{L}}(\cdot)$, like ${\mathcal{L}}({\mathfrak{H}})$. We set ${\mathbb{R}}_{+}:=[0,\infty)$. If a function is called measurable, then it means Lebesgue measurable. The notation “a.e.” means that a statement or relation fails at most for a set of Lebesgue measure zero. In the following we use the notation $\operatorname*{ess\,sup}_{(t,s)\in{\Delta}}$ or $\operatorname*{ess\,sup}_{(t,s)\in{\Delta}_{0}}$. In that case the Lebesgue measure of ${\mathbb{R}}^{2}$ is meant.

We point out that we call operator ${\mathcal{K}}$ to be generator of a semigroup $\{e^{-\tau{\mathcal{K}}}\}_{\tau\in{\mathbb{R}}_{+}}$, see e.g. ReedSim-II1975 , although in EngNag2000 ; Kato1980 it is the operator $-{\mathcal{K}}$, which is called the generator.

2 Evolution semigroups

Below we consider the Hilbert space ${\mathfrak{K}}=L^{2}({\mathcal{I}},{\mathfrak{H}})$ consisting of all measurable functions $f(\cdot):{\mathcal{I}}\longrightarrow{\mathfrak{H}}$ such that the norm function $\|f(\cdot)\|:{\mathcal{I}}\longrightarrow{\mathbb{R}}_{+}$ is square integrable. Further, let $D_{0}$ be the generator of the right-hand sift semigroup on $L^{2}({\mathcal{I}},{\mathfrak{H}})$, i.e.

[TABLE]

Notice that $e^{-{\tau}D_{0}}=0$ for ${\tau}\geq T$. The generator $D_{0}$ is given by

[TABLE]

We remark that $D_{0}$ is a closure of the maximal symmetric operator and its semigroup is contractive.

Further we consider the multiplication operator ${\mathcal{A}}$ in $L^{2}({\mathcal{I}},{\mathfrak{H}})$,

[TABLE]

If (S1) is satisfied, then ${\mathcal{A}}$ is self-adjoint and ${\mathcal{A}}\geq I_{L^{2}({\mathcal{I}},{\mathfrak{H}})}$. For the resolvent one has the representation

[TABLE]

and the corresponding semigroup $\{e^{-{\tau}{\mathcal{A}}}\}_{{\tau}\in{\mathbb{R}}_{+}}$ is given by

[TABLE]

Notice that the operators $e^{-{\tau}D_{0}}$ and $e^{-{\tau}{\mathcal{A}}}$ commute. Let us consider the contraction semigroup

[TABLE]

Obviously, the semigroup $\{{\mathcal{U}}_{0}(\tau)\}_{{\tau}\in{\mathbb{R}}_{+}}$ admits the representation (1.13). Due to the maximal $L^{2}$-regularity of $A$, cf. Arendt2007 , its generator ${\mathcal{K}}_{0}$ is given by

[TABLE]

Further we consider the multiplication operator ${\mathcal{B}}$, defined as

[TABLE]

If (S1) is satisfied, then ${\mathcal{B}}$ is self-adjoint and non-negative. For the resolvent we have the representation

[TABLE]

for a.e. $t\in{\mathcal{I}}$. The semigroup $\{e^{-{\tau}{\mathcal{B}}}\}_{{\tau}\in{\mathbb{R}}_{+}}$, admits the representation

[TABLE]

for a.e. $t\in{\mathcal{I}}$.

By (NeiStephZagr2016, , Proposition 4.4) we get that under the assumptions (S1) and (S2) the operator

[TABLE]

is a generator of a contraction semigroup on $L^{2}({\mathcal{I}},{\mathfrak{H}})$. From (NeiStephZagr2016, , Proposition 4.5) we obtain that ${\mathcal{K}}$ is the generator of an evolution semigroup. Because ${\mathcal{K}}$ is a generator of a contraction semigroup it turns out that the corresponding propagator consists of contractions.

If $\{U(t,s)\}_{(t,s)\in{\Delta}_{0}}$ is a propagator, then by virtue of (1.13) it defines a semigroup, which by definition is an evolution semigroup. It turns out that there is a one-to-one correspondence between the set of evolution semigroups on $L^{2}({\mathcal{I}},{\mathfrak{H}})$ and propagators. It is interesting to note that evolution generators can be characterize quite independent from a propagator, see (Nei1981, , Theorem 2.8) or (NeiStephZagr2016, , Theorem 3.3).

3 Results

We start with a general observation concerning the conditions (S1)-(S3).

Remark 3.1

If the conditions (S1)-(S3) are satisfied for some ${\alpha}\in[0,1)$, then they are also satisfied for each $\alpha^{\prime}\in({\alpha},1]$. Indeed, the condition (S1) is obviously satisfied. To show (S2) we note that $\mathrm{dom}(A^{\alpha^{\prime}})\subseteq\mathrm{dom}(A^{\alpha})\subseteq\mathrm{dom}(B(t))$ for a.e. $t\in{\mathcal{I}}$. Using the representation

[TABLE]

for a.e. $t\in{\mathcal{I}}$ we get that the map $B(\cdot)A^{-{\alpha^{\prime}}}:{\mathcal{I}}\longrightarrow{\mathcal{L}}({\mathfrak{H}})$ is strongly measurable. Further, from (3.1)

[TABLE]

Moreover we have

[TABLE]

$(t,s)\in{\mathcal{I}}\times{\mathcal{I}}$, which shows that there is a constant $L_{{\alpha^{\prime}},{\beta}}\leq L_{{\alpha},{\beta}}$ such that

[TABLE]

holds for $(t,s)\in{\mathcal{I}}\times{\mathcal{I}}$. $\triangle$ **

Since $A$ is self-adjoint and non-negative, one has $\|A^{\gamma}e^{-\tau A}\|\leq{1}/{\tau^{\gamma}}$ for any $\tau\in{\mathbb{R}}_{+}$ and $\gamma\in[0,1]$. Then by virtue of (2.2) and of (1.10), (2.3) one gets the estimates

[TABLE]

3.1 Auxiliary estimates

In this section we prove a series of estimates necessary to establish (1.12). The following lemma can be partially derived from (NeiStephZagr2016, , Lemma 7.4).

Lemma 3.2

Let the assumptions (S1) and (S2) be satisfied. Then for any $\gamma\in[{\alpha},1)$ there is a constants ${\Lambda}_{\gamma}\geq 1$ such that

[TABLE]

holds.

Proof

The proof of the first estimate follows from Lemma 7.4 of NeiStephZagr2016 and Remark 3.1. The second estimate can be proved similarly as the first one. One has only to modify the proof of Lemma 7.4 of NeiStephZagr2016 in a suitable manner and to apply again Remark 3.1. $\Box$

Remark 3.3

Lemma 2.1 of IchinoseTamura1998 claims that for the Lipschitz case the solution operator $\{U(t,s)\}_{(t,s)\in{\Delta}}$ of (1.1) admits the estimates

[TABLE]

for ${\gamma}\in[0,1]$. Proposition 2.1 of NeiStephZagr2018b immediately yields that the corresponding evolution semigroup $\{{\mathcal{U}}({\tau})=e^{-{\tau}{\mathcal{K}}}\}_{\tau\in{\mathbb{R}}_{+}}$ satisfies the estimates (3.3) for ${\gamma}=1$. $\triangle$ **

Now we set

[TABLE]

Notice that ${\mathcal{T}}({\tau})=0$ for ${\tau}\geq T$.

Lemma 3.4

Let the assumptions (S1) and (S2) be satisfied. Then for any ${\gamma}\in[{\alpha},1)$ the estimates

[TABLE]

hold for ${\tau}\geq 0$, where

[TABLE]

Proof

The proof of the first estimate follows from Lemma 7.6 of NeiStephZagr2016 and Remark 3.1. The specific constant $2C_{\gamma}$ is obtained following carefully the proof of Lemma 7.6 of NeiStephZagr2016 . The second estimate can be proved modifying the proof of the first estimate in an obvious manner. $\Box$

Lemma 3.5

Let the assumptions (S1)-(S3) be satisfied. Then for any ${\gamma}\in[{\alpha},1)$ and ${\beta}\in(0,1)$ there is a constant $Z_{{\gamma},{\beta}}>0$ such that

[TABLE]

holds where $\varkappa:=\min\{{\gamma},{\beta}\}$.

Proof

We use the representation:

[TABLE]

which yields

[TABLE]

Hence, we obtain the identity

[TABLE]

which leads to the estimate

[TABLE]

Note that by (3.2) and (3.6) one gets

[TABLE]

for ${\sigma}>0$. Due to (3.12) one estimates (3.8) as

[TABLE]

Since the fundamental properties of semigroups and (3.6) yield

[TABLE]

and

[TABLE]

we get for (3.8) the estimate

[TABLE]

To estimate (3.9) we recall that $\mathcal{A}$ and $\mathcal{K}_{0}$ commute. Then by (3.6) one gets

[TABLE]

where (3.13) was used for the last inequality.

To estimate (3.10) we have

[TABLE]

To estimate (3.11) we use the representation

[TABLE]

that yields

[TABLE]

Then by (3.6) and by semigroup properties one gets

[TABLE]

and

[TABLE]

The last term is obtained by using (S3) (for $\alpha$ substituted by $\gamma$) and the definitions (2.1), (2.4):

[TABLE]

Summing up one finds that

[TABLE]

Using the estimates (3.14), (3.15), (3.16) and (3.17) we get

[TABLE]

or returning back to its derivative

[TABLE]

Since

[TABLE]

we find the estimate

[TABLE]

which yields the estimate

[TABLE]

or after integration:

[TABLE]

If ${\gamma}\in[{\alpha},1)$ and ${\gamma}\leq{\beta}<1$, then one gets

[TABLE]

${\tau}\in{\mathbb{R}}_{+}$, which immediately yields (3.7).

If ${\gamma}\in[{\alpha},1)$ and $0<{\beta}<{\gamma}$, then one can rewrite it as

[TABLE]

${\tau}\in{\mathbb{R}}_{+}$, which shows (3.7) for this choice of $\gamma$ and $\beta$. $\Box$

Remark 3.6

For ${\gamma}\in[{\alpha},1)$ and ${\gamma}\leq{\beta}<1$ we find from (3.18) that

[TABLE]

For ${\gamma}\in[{\alpha},1)$ and $0<{\beta}<{\gamma}$ we get from (3.19) that

[TABLE]

Here $C_{\gamma}:=\operatorname*{ess\,sup}_{t\in{\mathcal{I}}}\|{\mathcal{B}}{\mathcal{A}}^{-{\gamma}}\|$, see (3.3), and $L_{{\gamma},{\beta}}$ is the Hölder constant of the function $A^{-{\gamma}}B(\cdot)A^{-{\gamma}}:{\mathcal{I}}\longrightarrow{\mathcal{L}}({\mathfrak{H}})$, see (S3). **

Lemma 3.7

Let the assumptions (S1) and (S2) be satisfied. Then

[TABLE]

for ${\gamma}\in[{\alpha},1)$.

Proof

We use the representation

[TABLE]

which yields

[TABLE]

Using the semigroup property we obtain for the first term the representation:

[TABLE]

Hence, by (3.3) and (3.6) one gets

[TABLE]

To estimate the second term we use the inequality

[TABLE]

Using (3.2) and (3.13) we estimate the second term as

[TABLE]

Now the estimates (3.22) and (3.23) yield (3.21). $\Box$

Lemma 3.8

Let the assumption (S1) be satisfied. If for each ${\gamma}\in[{\alpha},1)$ there is a constant $M_{\gamma}>0$ such that

[TABLE]

holds for ${\mathcal{T}}({\tau})$ defined in (3.4), then

[TABLE]

holds for ${\sigma}\in[0,{\gamma}]$ and ${\delta}:={{\sigma}}/{{\gamma}}$.

Proof

If (3.24) is satisfied, then

[TABLE]

holds, which is equivalent to

[TABLE]

or

[TABLE]

Let ${\delta}\in(0,1)$. Using the Heinz inequality (BirSolom1987, , Theorem X.4.2) we get

[TABLE]

Since ${\mathcal{T}}({\tau})^{m}({\mathcal{T}}({\tau})^{*})^{m}$ is a self-adjoint contraction we get

[TABLE]

which yields

[TABLE]

or

[TABLE]

Therefore, one gets

[TABLE]

or

[TABLE]

Setting ${\delta}={{\sigma}}/{{\gamma}}$ we obtain the proof of (3.25). $\Box$

Lemma 3.9

Let the assumptions (S1) and (S2) be satisfied and let ${\gamma}\in({\alpha},1)$. Then there is a constant $M_{\gamma}>0$ such that

[TABLE]

holds for any T>0\ if $\ {\tau}\in(0,\tfrac{T}{n})$ and $n\geq n_{0}$ where $n_{0}:=\lfloor(2(\frac{{\Lambda}_{\gamma}}{1-{\gamma}}+1)C_{\gamma})^{\tfrac{1}{1-{\gamma}}}T\rfloor+1$ and $\lfloor x\rfloor$ denotes the largest integer smaller than $x$.

Proof

Let $M_{\gamma}>0$ be a constant which satisfies the inequality

[TABLE]

for $n\geq n_{0}$. Here constants ${\Lambda}_{\gamma}$ and $C_{\gamma}$ are defined by Lemma 3.2 and Lemma 3.4, respectively, while $B(\cdot,\cdot)$ denotes the Euler Beta-function. (Note that such $M_{\gamma}>0$ always exists, see Remark 3.10 below.)

Let $m=1$. Then by (3.2) and (3.4) we get

[TABLE]

for ${\tau}>0$ and, in particular, for ${\tau}\in(0,{T}/{n})$. Hence (3.26) holds for $m=1$.

Let us assume that (3.26) holds for $l=1,2,\ldots,m-1$, with $m\leq n$, i.e.

[TABLE]

for ${\tau}\in(0,{T}/{n})$. We are going to show that (3.28) holds for $l=m$. To this aim we use the representation

[TABLE]

which implies

[TABLE]

Hence

[TABLE]

or

[TABLE]

for $m=3,4,\dots$. This yields the inequality

[TABLE]

for $m=3,4,\dots$. From Lemma 3.2 we get the estimates

[TABLE]

and consequently:

[TABLE]

Then summing up estimates for the first two terms in the right-hand side of (3.29) we obtain

[TABLE]

Next we get for the third term in the right-hand side of (3.29) the estimate

[TABLE]

$m=2,3,\dots$. Then using Lemma 3.7 we find that

[TABLE]

By assumption (3.28) this yields

[TABLE]

for ${\tau}\in(0,{T}/{n})$, which leads to

[TABLE]

Finally one gets for the sum in (3.29)

[TABLE]

Then by Lemma 3.2 this implies

[TABLE]

Taking into account Lemma 3.4 we get

[TABLE]

Finally, using assumption (3.28) and Lemma 3.8 one obtains

[TABLE]

or

[TABLE]

for ${\tau}\in(0,{T}/{n})$. Since Lemma 3.11 below yields

[TABLE]

where $B(\cdot,\cdot)$ is the Euler Beta-function, we get

[TABLE]

which in turn leads to

[TABLE]

for $m=2,3,\dots$ and any ${\tau}\in(0,{T}/{n})$.

Now we take into account (3.29), (3.30), (3.31) and (3.33) to conclude that

[TABLE]

for $m=2,3,\dots$ and ${\tau}\in(0,{T}/{n})$. Then

[TABLE]

¿From assumption (3.27) we get

[TABLE]

for $n\geq n_{0}$, which shows that (3.28) holds for $l=1,2,3,\dots,n$ and $n\geq n_{0}$ which proves (3.26). $\Box$

Remark 3.10

One checks that condition (3.27) is always satisfied for sufficiently large $M=M_{\gamma}$ and $n\geq n_{0}$. Indeed, after setting

[TABLE]

we get the condition

[TABLE]

which yields

[TABLE]

or

[TABLE]

Since $n>c^{\tfrac{1}{1-{\gamma}}}_{1}$ we have $1-{c_{1}}/{n^{1-{\gamma}}}>0$. The left-hand side tends to zero if $M\to\infty$. Hence, choosing $M$ sufficiently large we guarantee the existence of $M_{\gamma}$ such that condition (3.27) is satisfied for any $n\geq n_{0}$. $\triangle$ **

It remains only to verify the following statement.

Lemma 3.11

Let ${\alpha}\in[0,1)$ and ${\gamma}\in[{\alpha},1)$. Then

[TABLE]

the estimate holds where $B(\cdot,\cdot)$ is the Euler Beta-function.

[TABLE]

Proof

If $x\in(k-1,k]$, then

[TABLE]

for $k=1,2,\ldots,n-1$. Hence

[TABLE]

Therefore

[TABLE]

or

[TABLE]

$\Box$

3.2 Main Results

In this section we collect our main results and their proofs. They are based on preliminaries established in Section 3.1.

Theorem 3.12

Let the assumptions (S1) -(S3) be satisfied and let ${\beta}>2{\alpha}-1$. Then there is a constant $R_{\beta}>0$ such that

[TABLE]

holds for $n\in{\mathbb{N}}$ and $\tau\in{\mathbb{R}}_{+}$.

Proof

Taking into account the representation

[TABLE]

or, identically,

[TABLE]

we obtain the estimate

[TABLE]

$n=3,4,\ldots\,$.

Note that using Lemma 3.2 and Lemma 3.4 one gets

[TABLE]

which yields

[TABLE]

for $n=3,4,\ldots$ and $\tau\in[0,T]$.

Now using Lemma 3.4 and Lemma 3.9 for $m=n-1$ we find

[TABLE]

for $n\geq n_{0}$, where $n_{0}$ is defined in Lemma 3.9 and $\tau\in[0,T]$. Hence,

[TABLE]

Taking into account Lemma 3.2, Lemma 3.5 and Lemma 3.9 (for $\varkappa=\min\{\gamma,\beta\}$) one gets

[TABLE]

for $n>\max\left\{2,n_{0}\right\}$ and $\tau\in[0,T]$ . Then by (3.32) we obtain

[TABLE]

or

[TABLE]

Therefore, by virtue of (3.35), (3.36), (3.37) and (3.38) we get for $n>\max\left\{2,n_{0}\right\}$ and $\tau\in[0,T]$ the estimate

[TABLE]

If ${\alpha}<{\beta}<1$, then we choose ${\gamma}={\beta}$, i.e., $\varkappa={\beta}$ and $1+\varkappa-2{\gamma}=1-{\beta}\geq 0$. Setting

[TABLE]

one obtains the estimate

[TABLE]

for $n>\max\left\{2,n_{0}\right\}$ and $\tau\in[0,T]$ .

Now let $0<{\beta}\leq{\alpha}$. Since $1+{\beta}-2{\alpha}>0$, there exists ${\gamma}\in({\alpha},1)$ such that $1+{\beta}-2{\gamma}\geq 0$. Indeed, there is a $\varepsilon>0$ verifying $1+{\beta}-2{\alpha}>2\varepsilon$. Setting ${\gamma}={\alpha}+\varepsilon$ we get $1+{\beta}-2{\gamma}>0$. Notice that $\varkappa={\beta}$. Then setting

[TABLE]

we obtain (3.39) for $n>\max\left\{2,n_{0}\right\}$.

Both results immediately imply that there is a constant $R_{\gamma}$ such that (3.34) holds for $\tau\in[0,T]$ and $n\in{\mathbb{N}}$. Finally, using ${\mathcal{U}}(\tau)=0$ and ${\mathcal{T}}({\tau}/{n})^{n}=0$ for $\tau\geq T$ we obtain (3.28) for any $\tau\in{\mathbb{R}}_{+}$. $\Box$

Now we set

[TABLE]

Corollary 3.13

Let the assumptions (S1) -(S3) be satisfied and ${\beta}>2{\alpha}-1$. Then there exists ${{\widetilde{R}}}_{\beta}>0$ such that estimate

[TABLE]

holds for $n\in{\mathbb{N}}$ and $\tau\in{\mathbb{R}}_{+}$.

Proof

Notice that

[TABLE]

Hence

[TABLE]

which yields the estimate

[TABLE]

Obviously, one has

[TABLE]

Using

[TABLE]

we get the estimate

[TABLE]

Taking into account condition (S2) and Lemma 3.2 we find

[TABLE]

where we have used that $e^{-\tau{\mathcal{K}}}=0$ for $\tau\geq T$.

Further, we have

[TABLE]

$\tau\in{\mathbb{R}}_{+}$, $n\in{\mathbb{N}}$. Then using

[TABLE]

$\tau\in{\mathbb{R}}_{+}$, $n\in{\mathbb{N}}$, we find the estimate

[TABLE]

Applying again Lemma 3.2 one gets

[TABLE]

The insertion of (3.42) and (3.43) into (3.41) yields

[TABLE]

Then by Theorem 3.12 we obtain

[TABLE]

Therefore, by setting $R^{\prime}_{\gamma}:=2C_{\alpha}{\Lambda}_{\alpha}+R_{\gamma}$ we obtain

[TABLE]

which yields

[TABLE]

Let $\tau=\tau^{\prime}{n}/{(n+1)}$ for $\tau^{\prime}\in{\mathbb{R}}_{+}$. Then

[TABLE]

or

[TABLE]

$\tau\in{\mathbb{R}}_{+}$, $n\in{\mathbb{N}}$. Setting ${{\widetilde{R}}}_{\gamma}:=\max\{2,2^{\gamma}R^{\prime}_{\gamma}\}$ we prove (3.40).

$\Box$

These results can be immediately extended to propagators. To this end we set

[TABLE]

$t_{j}:=s+j\tfrac{t-s}{n}$, $j=0,1,2,\ldots,n$, in analogy to (1.6).

Theorem 3.14

Let the assumptions (S1)-(S3) be satisfied. Further, let $\{U(t,s)\}_{(t,s)\in{\Delta}_{0}}$ be the propagator corresponding to the evolution generator ${\mathcal{K}}$ and let $\{V_{n}(t,s)\}_{(t,s)\in{\Delta}_{0}}$ and $\{{{\widetilde{V}}}_{n}(t,s)\}_{(t,s)\in{\Delta}_{0}}$ be defined by (1.6) and (3.44), respectively. If ${\beta}>2{\alpha}-1$, then the estimates

[TABLE]

hold for $n\in{\mathbb{N}}$, where the constants $R_{\gamma}$ and ${{\widetilde{R}}}_{\gamma}$ are those of Theorem 3.12 and Corollary 3.13.

Proof

Note that Proposition 2.1 of NeiStephZagr2018b yields

[TABLE]

Then applying Theorem 3.12 we prove (3.45).

To proof the second estimate we use Proposition 3.8 of NeiStephZagr2018 where the relation

[TABLE]

was shown. Applying Corollary 3.13 we complete the proof. $\Box$

4 Example

As an example we consider the diffusion equation perturbed by a time-dependent scalar potential. For this aim let ${\mathfrak{H}}=L^{2}(\Omega)$, where $\Omega\subset\mathbb{R}^{3}$ is a bounded domain with sufficiently smooth boundary. Domains in higher dimension can be treated analogously. The equation reads as

[TABLE]

where $\Delta$ denotes the Laplace operator in $L^{2}(\Omega)$ with Dirichlet boundary conditions, i.e. $\Delta:\mathrm{dom}(\Delta)=H^{2}(\Omega)\cap H^{1}_{0}(\Omega)\rightarrow L^{2}(\Omega)$ and $H^{1}_{0}(\Omega)$ denotes the subset of functions that vanish at the boundary. Then operator $-\Delta$ is self-adjoint on ${\mathfrak{H}}$ and positive. For any $\alpha\in(0,1)$ the fractional power of operator $-\Delta$ is defined on the domain $\mathrm{dom}((-\Delta)^{\alpha})$, i.e. $(-\Delta)^{\alpha}:\mathrm{dom}((-\Delta)^{\alpha})\rightarrow L^{2}(\Omega)$. The domain is given by a fractional Sobolev space and for $\alpha>1/2$, we have $\mathrm{dom}((-\Delta)^{\alpha})=H_{0}^{2\alpha}(\Omega)\subset H^{2\alpha}(\Omega)$ (see LionsMagenes1972 for more information).

Moreover let $B(t)$ denote a time-dependent scalar-valued multiplication operator given by

[TABLE]

where $V:{\mathcal{I}}\times\Omega\rightarrow\mathbb{R}$ is measurable. We assume that the potential $V(\cdot,\cdot)$ is real and non-negative. Then $B(t)$ is obviously self-adjoint and non-negative on ${\mathfrak{H}}$.

Theorem 4.1

Let $A$ be the Laplacian operator $-{\Delta}$ with Dirichlet boundary conditions in $L^{2}(\Omega)$, see above. Further, let $\{B(t)\}_{i\in{\mathcal{I}}}$ be the family of multiplication operators defined by (4.2). If $V(\cdot,\cdot):{\mathcal{I}}\times\Omega\longrightarrow{\mathbb{R}}$ is measurable, real, non-negative with regularity $V\in L^{\infty}({\mathcal{I}},L^{2+\varepsilon}(\Omega))\cap C^{\beta}({\mathcal{I}},L^{1+\varepsilon}(\Omega))$ for ${\beta}\in(0,1)$ and some $\varepsilon>0$, then the assumptions (S1)-(S3) are satisfied with $\alpha\in[3/4,1)$. Moreover, if ${\beta}>2{\alpha}-1$ then the converging rates of Theorem 3.12, Corollary 3.13 and Theorem 3.14 hold.

Proof

Since $\Omega$ is bounded there one has $\inf{\sigma}(A)>0$ which does not satisfy $A\geq I$ in general and, hence, assumption (S1) is not satisfied. Nevertheless $\inf{\sigma}(A)>0$ is sufficient to prove the converging results. So we can believe that (S1) is satisfied.

Let $\alpha\geq 3/4$. Using the Sobolev space embeddings, we get that $H^{2\alpha}(\Omega)\subset L^{\gamma}(\Omega)$ for any $\gamma\in[2,\infty[$. Hence, if $V\in L^{\infty}({\mathcal{I}},L^{2+\varepsilon}(\Omega))$, we conclude that the function $[0,T]\ni t\mapsto B(t)(-\Delta)^{-\alpha}$ is essentially operator-norm bounded in $t\in{\mathcal{I}}$ and thus, (S2) is satisfied. Now, we consider

[TABLE]

The function $F(\cdot):{\mathcal{I}}\rightarrow\cal L({\mathfrak{H}})$ is bounded for fixed $t\in[0,T]$ if for any $f,g\in H^{2\alpha}(\Omega)$ the function $\langle f,B(t)g\rangle$ is bounded. This holds since $V(t,\cdot)\in L^{1+\varepsilon}(\Omega)$ and $H^{2\alpha}(\Omega)\subset L^{\gamma}(\Omega)$ for any $\gamma\in[2,\infty[$. Hence we conclude that (S3) is satisfied and the claim is proved. $\Box$

Theorem 4.1 provides a convergence rate of an approximation of the solution of (4.1) by the time-ordered product

[TABLE]

This looks elaborate, but is indeed simple. There are strategies to compute the semigroup of the Laplace operator for bounded domains and there are also explicit formulas on special domains like disks etc. The factors $e^{-\tau V(t_{j})}$, $j=1,2,\ldots,n$ are scalar valued and can be easily computed.

Acknowledgment

We thank Takashi Ichinose and Hideo Tamura for the explanation of details of the proof of Theorem 1.1 of IchinoseTamura1998 , which makes possible to prove Lemma 3.8 and Lemma 3.9.