STUDIA MATHEMATICA
Online First version

Discrete maximal regularity for abstract Cauchy problemsbyTomoya Kemmochi (Tokyo)
Abstract. Maximal regularity is a fundamental concept in the theory of nonlinear
partial differential equations, for example, quasilinear parabolic equations, and the Navier–Stokes equations. It is thus natural to ask whether the discrete analogue of this notion holds
when the equation is discretized for numerical computation. In this paper, we introduce
the notion of discrete maximal regularity for the finite difference method (θ-method),and show that discrete maximal regularity is roughly equivalent to (continuous) maximal
regularity for bounded operators in the case of UMD spaces. The feature of our result isthat it includes the conditionally stable case (0 ≤ θ < 1/2). We pay close attention to
the dependence of the constants appearing in estimates. In addition, we show that thischaracterization is also true for unbounded operators in the case of the backward Euler
method.

1. Introduction. In this paper, we consider the following abstract
Cauchy problem in a Banach space X: 0u (t) = Au(t) + f (t), t > 0,u(0) = 0,
where u is an unknown X-valued function, f is a given one, and A is a linear
operator on X. The operator A is said to have (continuous) maximal regularity if, for some p ∈ (1, ∞) and every f ∈ Lp (0, ∞; X), the above problem
has a unique solution u (the precise meaning of “solution” is described in
Definition 2.4) satisfying
ku0 kLp (0,∞;X) + kAukLp (0,∞;X) ≤ Ckf kLp (0,∞;X) ,uniformly with respect to f . For example, it is known that the Laplace
operator and the Stokes operator have maximal regularity under suitableconditions, and that this property can be applied to quasilinear parabolic
equations and the Navier–Stokes equations (see e.g. [2, 26]). We are con2010 Mathematics Subject Classification: Primary 65J08; Secondary 65M06.
Key words and phrases: discrete maximal regularity, maximal regularity, abstract Cauchy
problem, finite difference method, Fourier multiplier theorem.
Received 9 January 2016; revised 29 June 2016.
Published online 9 September 2016.
DOI: 10.4064/sm8495-7-2016[1]

c Instytut Matematyczny PAN, 2016

2

T. Kemmochi

cerned about the numerical computation of the Cauchy problem above. As
is well-known, the analytic semigroup theory and its discrete counterparts
play important roles in construction and study of numerical schemes forparabolic equations (cf. [12, 13, 14, 24, 25]). Hence, it is natural to ask
whether a discrete analogue of maximal regularity holds when the above
problem is discretized for numerical computations. Moreover, if this is thecase, then it is expected that the discrete version of maximal regularity canbe applied to the numerical analysis of nonlinear evolution equations, forexample, the stability analysis and the error estimate of the finite element
approximation for the equations as given above. Indeed, Geissert [15, 16]
and Li [20] considered the continuous maximal regularity for the discrete
Laplacian, and applied it to the semidiscrete problem for the linear andsemilinear heat equations. However, since they only dealt with the semidiscrete problem, the results cannot be applied to the analysis of practicalcomputations. Thus, we need to consider the time-discretized problem andthe discrete version of maximal regularity.
In the present paper, we concentrate on the discretization of the time
variable, and postpone that of the space variables to further studies (cf. [17]).
That is, we discretize the Cauchy problem by the finite difference method:
( n+1
u
− un
= Aun+θ + f n+θ , n ∈ N = {0, 1, 2, . . . },(1.1)τ
u0 = 0,where τ > 0 is the time step, θ ∈ [0, 1] is a fixed parameter, u = (un ) isan unknown X-valued sequence, and f = (f n ) is a given one. We say that
A has discrete maximal regularity if for every f ∈ lp (N; X), there exists a
unique solution u of (1.1) with the estimatekDτ uklτp (N;X) + kAuθ klτp (N;X) ≤ Ckfθ klτp (N;X) ,uniformly with respect to τ . The meaning of symbols in the above inequality
will be specified in Section 2. We are interested in (1) when the operator A
has discrete maximal regularity, and (2) what the constant C depends on.
We now summarize several previous studies. For more details, one can refer to [1]. The first one is Blunck’s result [5]. Here the forward Euler method(θ = 0) was considered, and the notion of discrete maximal regularity was
introduced. The main result was the discrete version of the operator-valued
Fourier multiplier theorem (cf. [27, 21]) and a characterization of discrete
maximal regularity. While these results seem powerful, Blunck only considered the case where the time step is unity (i.e., τ = 1). Therefore, theresult, especially the characterization of discrete maximal regularity, cannot be directly applied to numerical analysis. Although there are several
works following Blunck such as [9, 23], these results are not applicable toanalysis of numerical schemes either. Another study we mention is that of

Discrete maximal regularity

3

Ashyralyev et al. [3]. They considered the backward Euler method (θ = 1)
and the Crank–Nicolson method (θ = 1/2), with an arbitrary time step
τ > 0, and showed that if A has continuous maximal regularity, then it also
has discrete maximal regularity for θ = 0, 1/2. Recently, Kovács et al. [18]showed the same results for A-stable time discretizations, such as BDF andimplicit Runge–Kutta methods. They also considered fully-discretized problems with abstract space-discretization. However, the constants appearing
in those results may depend on the Banach spaces under consideration, due
to utilization of Blunck’s discrete multiplier theorem. In contrast to thoseresults, we focus on general single-step finite difference methods (1.1) including the conditionally stable case, and we shall pay attention to what
the constants depend on, especially whether they depend on Banach spaces.
Our main goal is to show that continuous maximal regularity implies discrete maximal regularity for general θ-methods in the case of UMD spaces.
When θ ∈ [1/2, 1], our main theorem (Theorem 3.2) includes the results of
Ashyralyev et al., and the proof is integrated and simplified. Our result also
includes the conditionally stable case (θ ∈ [0, 1/2)), which is not consideredin the literature and is the main feature of the present paper. In this case,a certain stability condition is assumed, which is not mentioned in [5] when
τ = 1. This condition is reasonable since it often appears in the context ofnumerics for parabolic problems. Many operators exist that have continuous
maximal regularity. Furthermore, many approaches to continuous maximal
regularity have already been developed. Therefore, even if we do not knowwhether a given operator has maximal regularity, this can be investigated.
As a result, our sufficient condition is quite reasonable from both analyticaland numerical viewpoints.
As in the previous studies, our method is based on R-boundedness andBlunck’s discrete Fourier multiplier theorem (Theorem 2.11). However, as
mentioned before, the constant appearing in this theorem depends on the
Banach space X. This is troublesome in view of numerical analysis, since
X is assumed to be a finite-dimensional space that depends on the space
discretization parameter (e.g., En in [3] and Xh in [18]) in applications.
Therefore, we discuss this problem, and obtain an applicable version of themain result (Corollary 3.3). Although one may think that this discussion is
obvious, it is important from the numerical point of view.
We also consider the opposite assertion: discrete maximal regularity implies continuous maximal regularity (Theorem 3.5), of just mathematicalinterest. This question was considered in [5] when θ = 0 and τ = 1. In [5,Theorem 1.1], the power-boundedness for T = I + A is assumed, whichis equivalent to the stability condition mentioned above for τ = 1. However, it does not hold in general settings of numerical analysis. Therefore,
we present a numerical-analytic statement, that is, we show that a uniform

4

T. Kemmochi

estimate (for small τ ) implies continuous maximal regularity. This is provedfor general θ by a limiting argument as τ ↓ 0. See also Theorem 4.1.
As mentioned above, we restrict our consideration to the (time-)discrete
Cauchy problem. We have succeeded in applying the results of this paper
to the stability and convergence analysis of the finite element method forlinear and semilinear heat equations [17].
The plan of the rest of this paper is as follows. Section 2 is devoted topreliminaries on maximal regularity. In Subsection 2.1, we introduce thenotion of R-boundedness. It plays an important role in operator-valued
Fourier multiplier theorems, in both the continuous and discrete cases. Subsections 2.2 and 2.3 are devoted to continuous and discrete maximal regularity. In Subsection 2.3, we reduce the problem of discrete maximal regularity
to R-boundedness of certain sets of operators. Our main result appears in
Section 3. Its significance is in the dependence of the constant. We also
demonstrate that the converse of the main theorem holds.
We conclude this paper by dealing with some additional topics in Section 4, where we focus on the backward Euler method. In this case, we canalso show other analogous properties. This section consists of two parts.
First, we consider the characterization of discrete maximal regularity for
unbounded operators (Subsection 4.1). This corresponds to the result given
by Blunck, which deals with bounded operators. The key point is Yosida approximation. Next, we obtain an a priori estimate for non-zero initial values(Subsection 4.2). The results in this subsection are important for numericalanalysis of nonlinear equations. However, since the arguments are essentiallythe same as in [4, Chapter 2], which deals with the same problem in the timeinterval (0, 1), we omit the detailed proofs.
2. Preliminaries
2.1. R-boundedness. In this subsection, we introduce the definition of
R-boundedness and its basic property to be used later. For standard facts on
R-boundedness, we refer to [19, Section 2]. R-boundedness is a fundamental
concept in this paper since it plays a crucial role in Weis’s operator-valued
Fourier multiplier theorem on UMD spaces [27, Theorem 3.4], as well as inits discrete version [5, Theorem 1.3].
Definition 2.1. A Banach space X is an UMD space if for some p ∈(1, ∞) and C > 0, we have
n
X
(2.1)
u0 +
εj (uj − uj−1 ) p≤ Ckun kLp (Ω,F ,P ;X)
j=1

L (Ω,F ,P ;X)

for all n ∈ N, εj ∈ {±1}, and X-valued martingales (uj ) on a probability
space (Ω, F, P ).

Discrete maximal regularity

5

We refer to [7, 8, 6] for characterizations of UMD spaces by ζ-convexity
and the Hilbert transform.
Let X and Y be Banach spaces and let L(X, Y ) denote the space ofbounded operators from X to Y . Let {rj }j∈N be a sequence of independentand symmetric {±1}-valued random variables on [0, 1], for example, theRademacher functions rj (t) = sign[sin(2j+1 πt)].
Definition 2.2. A set T ⊂ L(X, Y ) is said to be R-bounded if there
exists a constant C > 0 such that
1 X
1 X
n
n
(2.2)
rj (t)Tj xj dt ≤ C
rj (t)xj
dt
0

j=0

Y

0

j=0

X

for all n ∈ N, x0 , . . . , xn ∈ X, and T0 , . . . , Tn ∈ T . The infimum of the C’s
satisfying (2.2) is called the R-bound of T and is denoted by R(T ).
We now introduce a property of sectorial operators. The following lemma
is a modification of [5, Corollary 3.5] and is obtained by the same argument
as in [5]. Here Σδ = {z ∈ C \ {0} | |arg z| < δ} is a sector domain in C forδ ∈ (0, π).
Lemma 2.3. Let A be a closed and densely defined linear operator on X.
Assume that there exists δ ∈ (0, π/2) such that Σπ/2+δ ⊂ ρ(A), and set
Tθ = {λR(λ; A) | λ ∈ Σθ } for θ ∈ (0, π/2 + δ). If Tπ/2 is R-bounded, thenso is Tπ/2+δ0 for each δ0 satisfying
1
0 < δ0 < min δ, arctan.
R(Tπ/2 )Moreover, R(Tπ/2+δ0 ) ≤ P1 (R(Tπ/2 )), where


2
X
(2.3)
P1 (X) =
1+
+X
1−α
α

is a polynomial of degree one with α = R(Tπ/2 ) tan δ0 .
2.2. Maximal regularity. We consider the following abstract Cauchy
problem in a Banach space X: 0u (t) = Au(t) + f (t), t > 0,
(2.4)
u(0) = 0,
where f : R+ = (0, ∞) → X is a given function, u : R+ → X is the unknown,
and A is a linear operator on X with domain D(A).
Definition 2.4. Let p ∈ (1, ∞). A linear operator A has maximal Lp regularity with a constant C if, for every f ∈ Lp (R+ ; X), (2.4) has a uniquesolution u with the following properties:

6

T. Kemmochi(1) u(t) ∈ D(A) for almost every t > 0,(2) u is strongly differentiable in X for almost every t > 0,(3) there exists a constant C > 0, independent of f , such that
(2.5)

ku0 kLp (R+ ;X) + kAukLp (R+ ;X) ≤ Ckf kLp (R+ ;X) .

In Definition 2.4, we do not require that u ∈ Lp (R+ ; X). However, if
0 ∈ ρ(A), maximal Lp -regularity implies that u ∈ Lp (R+ ; X), since kA · k isa norm in D(A) equivalent to the graph norm. Since maximal Lp -regularity
is p-independent (see e.g. [10, Theorem 4.2]), we say that A has maximal
regularity if A has maximal Lp -regularity for some p ∈ (1, ∞). To distinguish this from the discrete case below, we occasionally call this continuous
maximal regularity.
It is known that an operator of maximal regularity generates a bounded
analytic semigroup (see e.g. [10, Theorem 2.1]). Thus maximal regularity isstronger than analyticity of the semigroup. The Laplace operator and the
Stokes operator have maximal regularity under suitable conditions, which
can be applied to the analysis of quasilinear parabolic equations and theNavier–Stokes equations.
As a sufficient condition for maximal regularity, the result of Dore andVenni [11] is well-known. On the other hand, in [27, Corollary 4.4], Weischaracterized maximal regularity by the R-boundedness of some sets of operators.
Theorem 2.5 (Weis). Let X be a UMD space and T (t) a bounded analytic semigroup on X with generator A. Then the following conditions areequivalent:(a) A has maximal regularity.
(b) {λR(λ; A) | λ ∈ iR \ {0}} is R-bounded.
(c) {T (t) | t > 0} and {tAT (t) | t > 0} are R-bounded.

Remark 2.6. Let p ∈ (1, ∞), and let X and A be as in Theorem 2.5.
Assume that A has maximal Lp -regularity with a constant CMR . Then,
tracing the constants in [5, Proposition 1.4], one can observe thatR({λR(λ; A) | λ ∈ iR \ {0}}) ≤ cp CMR ,where cp > 0 is a constant depending only on p.

2.3. Discrete maximal regularity. We next discretize the notion of
maximal regularity. First, we need to consider the discrete problem for (2.4).
In this paper, we use the single-step finite difference method to discretize
the time variable. That is, we consider the discrete Cauchy problem in X: un+1 − un
= Aun+θ + f n+θ , n ∈ N,
(2.6)
τ
 0
u = 0,

Discrete maximal regularity

7

where τ > 0 is the time step, θ ∈ [0, 1] is a fixed parameter, f = (f n )n ∈ X N
is a given sequence, u = (un )n ∈ X N is an unknown sequence, andv n+θ = (1 − θ)v n + θv n+1

for v = (v n ) ∈ X N . Hereafter, we assume that A is bounded, since theproblem (2.6) is ill-posed for general f when A is unbounded and θ 6= 1.
In general, the discretization (2.6) is called the θ-method. It is known as
the forward Euler method when θ = 0, the backward Euler method when
θ = 1, and the Crank–Nicolson method when θ = 1/2. Note that the solvability of (2.6) is equivalent to the invertibility of I − θτ A, since (2.6) canbe rewritten as(2.7)(I − θτ A)un+1 = (I + (1 − θ)τ A)un + τ f n+θ .

In particular, if (2.6) is solvable, then the solution must be unique.
For the space X N , we introduce some notation.
Definition 2.7. Let p ∈ (1, ∞).

(1) We define the discrete Lp -norm k · klτp (N;X) as
kvklτp (N;X) =

∞
X

n=0

kv n kpX τ

1/pfor v = (v n ) ∈ lp (N; X).

(2) For v = (v n ) ∈ X N , τ > 0, and θ ∈ [0, 1], we define the sequencesDτ v, Av, and vθ as

v n+1 − v n, (Av)n = Av n , (vθ )n = v n+θ .
τNow, we can define the discrete version of maximal Lp -regularity.
(Dτ v)n =Definition 2.8. Suppose that p ∈ (1, ∞) and θ ∈ [0, 1]. A linear operator A has maximal lp -regularity with a constant C if, for every τ > 0 smallenough and f ∈ lp (N; X), problem (2.6) has a unique solution u = (un ) ∈ X N
satisfying(2.8)

kDτ uklτp (N;X) + kAuθ klτp (N;X) ≤ Ckfθ klτp (N;X) ,where C > 0 is independent of τ > 0 and f . We say that A has discrete
maximal regularity if A has maximal lp -regularity for some p ∈ (1, ∞).

We characterize maximal lp -regularity by the boundedness of the Fourier
multiplier. Hereafter, we write T = {z ∈ C | |z| = 1}, and we assume that
A is the infinitesimal generator of a bounded analytic semigroup on X, sothat (2.6) is solvable. When T \ {1} ⊂ ρ(Tτ ), we set(2.9)(2.10)Mτ (z) = (I − θτ A)−1 (z − 1)R(z; Tτ ),n(TMτ f ) = [F−1n(Mτ Ff )] ,z ∈ T,f ∈ c00 (Z; X), n ∈ Z,

8

T. Kemmochi

where(2.11)Tτ = (I − θτ A)−1 (I + (1 − θ)τ A).

Here and hereafter, F and F −1 are the Fourier transforms on Z and on T,
respectively, and c00 (Z; X) ⊂ X Z is the space of X-valued sequences withcompact supports.
Lemma 2.9. Let p ∈ (1, ∞), and let A ∈ L(X) be the infinitesimal
generator of a bounded analytic semigroup on X. Suppose that T \ {1} ⊂ρ(Tτ ). Then the following assertions are equivalent:(a) A has maximal lp -regularity.
(b) TMτ can be extended to a bounded operator on lp (Z; X), and its operator norm is bounded by a constant independent of τ > 0.
Proof. From (2.7), Dτ u is written as
n−1
i
h
X
n
−1
(Tτ − I)Tτn−j−1 f j+θ + f n+θ
(Dτ u) = (I − θτ A)
j=0

for n ∈ N. Therefore, by a basic computation, we can obtain(Dτ u)n = (TM f˜θ )n , n ∈ N,τwhere f˜θ is the zero extension of fθ to Z. Hence, we obtain the desired
equivalence.
Now, we check when T \ {1} ⊂ ρ(Tτ ). We defineζ −1(2.12)gθ,τ (ζ) =, ζ ∈ C.
θτ ζ + (1 − θ)τ

Assume that gθ,τ (T\{1}) ⊂ ρ(A), and let λ ∈ T\{1}. If θ = 1/2 and λ = −1,
then λI−Tτ = −4(2−τ A)−1 is invertible. Otherwise, since θτ λ+(1−θ)τ 6= 0,
we have(2.13)λI − Tτ = [θτ λ + (1 − θ)τ ](gθ,τ (λ)I − A)(I − θτ A)−1 ,which implies that λ ∈ ρ(Tτ ). It remains to determine the set gθ,τ (T \ {1}).
By a simple calculation,


 
−1
1



C
;
\ {0}, 0 ≤ θ < 1/2,
(1 − 2θ)τ (1 − 2θ)τg (T \ {1}) =


θ,τ

1
1
(2.14)

C;\ {0}, 1/2 < θ ≤ 1,(2θ − 1)τ (2θ − 1)τ

g (T \ {±1}) = iR \ {0}, θ = 1/2,
θ,τwhere C(a; r) = {z ∈ C | |z − a| = r} for a ∈ C and r > 0. Since A generates
a bounded analytic semigroup, we have gθ,τ (T \ {1}) ⊂ Σπ/2 \ {0} ⊂ ρ(A)when 1/2 < θ ≤ 1 (Σπ/2 is the closure in C). Therefore, we need no condition

Discrete maximal regularity

9for T \ {1} ⊂ ρ(Tτ ) in this case. Much the same is true when θ = 1/2.
However, an additional condition is necessary when 0 ≤ θ < 1/2. We thengive the following condition (S) (cf. Fig. 1).
CIm

θ = 1/2(imaginary axis)

gθ,τ

T
0

Im

σ(A)
1 Re

1
(2θ−1)τ

0−1(1−2θ)τ

0 ≤ θ < 1/2CRe

1/2 < θ ≤ 1
H \ {0} ⊂ ρ(A)Fig. 1. The set gθ,τ (T \ {1}) and condition (S)(S) The operator A satisfies−11
σ(A) ⊂ D∪ {0}.
;(1 − 2θ)τ (1 − 2θ)τHere, D(a; r) = {z ∈ C | |z − a| < r} is an open disk for a ∈ C andr > 0. Note that (S) is satisfied if τ is sufficiently small, since the spectrum of A ∈ L(X) is a bounded set. Now, we have a sufficient condition for
T \ {1} ⊂ ρ(Tτ ).
Lemma 2.10. Let θ ∈ [0, 1]. Assume that A ∈ L(X) is the infinitesimal
generator of a bounded analytic semigroup on X. Suppose that condition (S)is fulfilled when 0 ≤ θ < 1/2. Then T \ {1} ⊂ ρ(Tτ ). In particular, the two
assertions in the previous lemma are equivalent.
From the viewpoint of Lemma 2.9, we need to examine the boundednessof the multiplier operator. For this purpose, Blunck proved the following
multiplier theorem [5, Theorem 1.3]. This is the discrete version of Weis’s
operator-valued Fourier multiplier theorem [27, Theorem 3.4]. See also [21]for the scalar-valued discrete Fourier multiplier theorem.
Theorem 2.11 (Blunck). Let X be a UMD space, J = (−π, π) \ {0},and M : J → L(X). Set(2.15)

TM f = F −1 [M̃ Ff ],f ∈ c00 (Z; X),

where M̃ (z) = M (arg z) for z ∈ T \ {1}. Assume that M is differentiable
and the set(2.16)

T = {M (t) | t ∈ J} ∪ {(eit − 1)(eit + 1)M 0 (t) | t ∈ J}

10

T. Kemmochi

is R-bounded. Then TM can be extended to a bounded operator on lp (Z; X)
for all p ∈ (1, ∞). Moreover,(2.17)

kTM kL(lp (Z;X)) ≤ Cmul R(T ),where Cmul > 0 depends only on p and X.

In view of numerical analysis, it is troublesome that the constant Cmul
depends on the Banach space X, since X is supposed to be the finite element
space, which depends on the discretization parameter. Tracing the constantsin the proof of [5, Theorem 1.3], we find that the dependence on X is causedby the constant of the UMD property (2.1). Let CUMD (p, X) be the infimumof the constant C in (2.1). Then it is obvious that Y ⊂ X ⇒ CUMD (p, Y ) ≤CUMD (p, X). Therefore, we can state the following.
Corollary 2.12. Let X be a UMD space, and X0 ⊂ X a closed subspace. Furthermore, let J = (−π, π) \ {0} and M : J → L(X0 ). Define TM
as in (2.15) for f ∈ c00 (Z; X0 ), and T as in (2.16). Assume that M isdifferentiable and T is R-bounded. Then TM can be extended to a bounded
operator on lp (Z; X0 ) for all p ∈ (1, ∞). Moreover,(2.18)

kTM kL(lp (Z;X0 )) ≤ CR(T ),where C > 0 depends only on p and X, but is independent of X0 .
3. Main result. Our main result is based on the characterization given
in Lemma 2.9, with condition (S) assumed when θ ∈ [0, 1/2). However, toobtain a uniform estimate for τ , condition (S) is not sufficient. Therefore,
we consider the stronger condition given below (cf. Fig. 2).
π ⊂ ρ(A) Im
C
Σδ+
Σ
δ+π
22

r(A)
rε ≤

δ

−r

0

Re

gθ,τ (T)2r sin δ
Fig. 2. Condition (NR)δ,ε . Here, we set r = 1/[(1 − 2θ)τ ].

Discrete maximal regularity

11(NR)δ,ε The following two conditions are fulfilled:(NR1) S(A) ⊂ C \ Σδ+π/2 ,(NR2) (1 − 2θ)τ r(A) + ε ≤ 2 sin δ.

Here, the set

S(A) = {hx∗ , Axi | x ∈ X, x∗ ∈ X ∗ , kxk = kx∗ k = hx∗ , xi = 1}

is the numerical range of A, and r(A) = supz∈S(A) |z| (not the spectral
radius of A).
Remark 3.1. (1) One can observe that condition (NR)δ,ε is stronger
than (S), since σ(A) ⊂ S(A), where the overline indicates closure in C. See,for example, [22, Theorem 3.9 in Chapter 1].
(2) If A is the infinitesimal generator of a bounded analytic semigroupon X, then condition (NR1) is fulfilled for some δ ∈ (0, π/2). Therefore, if
one wants to achieve (NR)δ,ε , it suffices to fix ε small enough, and considerτ satisfying
2 sin δ − ετ≤.
(1 − 2θ)r(A)Now, we are in a position to state our main theorem.

Theorem 3.2 (Discrete maximal regularity for the θ-method). Let X
be a UMD space, and let p ∈ (1, ∞) and θ ∈ [0, 1]. Assume that A ∈ L(X)
has maximal Lp -regularity with a constant CA . When θ ∈ [0, 1/2), supposethat A satisfies condition (NR)δ,ε for some δ ∈ (0, π/2) and ε > 0. Then A
has maximal lp -regularity with a constant CDMR which depends only on p,
θ, CA , δ, ε, and the Banach space X. Moreover,(3.1)

CDMR = Cmul C0 ,where Cmul is the constant appearing in (2.17), and C0 > 0 depends only onp, θ, CA , δ, ε, but is independent of τ , X and the operator A.
Proof. The proof will be divided into four steps.
Step 1. Let Mτ , TMτ , and Tτ be defined by (2.9)–(2.11), respectively.
In view of Lemma 2.9, it suffices to show that TMτ is bounded in lp (Z; X).
Let J = (−π, π) \ {0}, and define M̌τ : J → L(X) asM̌τ (t) = Mτ (eit ) = (I − θτ A)−1 (eit − 1)R(eit ; Tτ ),t ∈ J.

By Theorem 2.11, we only need to show that the set
Tτ = {M̌τ (t) | t ∈ J} ∪ {(eit − 1)(eit + 1)M̌τ0 (t) | t ∈ J}
is R-bounded uniformly in τ > 0. We will calculate the sets
T1 = {M̌τ (t) | t ∈ J},

T2 = {(eit − 1)(eit + 1)M̌τ0 (t) | t ∈ J}.

12

T. Kemmochi

Let λ = eit for t ∈ J. Then, from (2.13), we have1
R(λ; Tτ ) =(I − θτ A)R(gθ,τ (λ); A),θτ λ + (1 − θ)τ

where gθ,τ is defined in (2.12). Therefore, setting µ = gθ,τ (λ), we have
(3.2)

M̌τ (t) = µR(µ; A),

which implies that
T1 = {µR(µ; A) | µ ∈ gθ,τ (T \ {1})}.

(3.3)

Let us now calculate T2 . Since
we haveM̌τ0 (t) = (I − θτ A)−1 ieit R(eit ; Tτ )[I − (eit − 1)R(eit ; Tτ )],(eit − 1)(eit + 1)M̌τ0 (t) = ieit (eit + 1)M̌τ (t)[I − (I − θτ A)M̌τ (t)].

Moreover, by (3.2),

I − (I − θτ A)M̌τ (t) = I − (I − θτ A)µR(µ; A) = (1 − θτ µ)[I − µR(µ; A)],

where µ = gθ,τ (eit ). Therefore,(eit − 1)(eit + 1)M̌τ0 (t) = ieit (eit + 1)(1 − θτ µ)M̌τ (t)[I − M̌τ (t)].

Noting that(z + 1)(1 − θτ gθ,τ (z)) ∈ C(1; 1)

for z ∈ T, regardless of θ and τ , we can obtainand thus(3.4)

R(T2 ) ≤ 4R(T1 )(1 + R(T1 )),R(Tτ ) ≤ 5R(T1 )(1 + R(T1 )),provided that T1 is R-bounded. Here, we have used the standard facts
R(S ∪ T ) ≤ R(S) + R(T ) and R(ST ) ≤ R(S)R(T ) (cf. [19, Fact 2.8]).
Hence, it suffices to prove the R-boundedness of T1 . Set
T0 = {isR(is; A) | s ∈ R \ {0}},which is R-bounded with
(3.5)

R(T0 ) ≤ cp CA

by the maximal Lp -regularity of A, where cp > 0 depends only on p (Remark 2.6).
Step 2. We prove the assertion when 1/2 ≤ θ ≤ 1. In this case, from
(3.3), (2.14), Theorem 2.5, and the convexity property of R-boundedness[19, Example 2.15], T1 is R-bounded with
(3.6)

R(T1 ) ≤ R(T0 ),Discrete maximal regularity

13

which is a uniform estimate in τ . Thus A has maximal lp -regularity and
CDMR ≤ Cmul R(Tτ ) ≤ Cmul · 5cp CA (1 + cp CA ),from (3.4)–(3.6), which implies (3.1) with

C0 = 5cp CA (1 + cp CA ).
Step 3. We assume that 0 ≤ θ < 1/2. In this step, we show that R(T1 )is bounded from above uniformly in τ . This case is not as simple as Step 2.
We set1−1γ = γθ,τ = gθ,τ (T) = C;,(1 − 2θ)τ (1 − 2θ)τand γ̇ = γ \ {0}. Then T1 = {µR(µ; A) | µ ∈ γ̇}. Take δ0 ∈ (0, δ) satisfying
1.
0 < δ0 < arctan
R(T0 )
Im C
δ0
Γ2,1
Γ2

Γ1

δ1

Γ2,2
Γ
2,2

−r

0

Re

δ
γ

Fig. 3. The arcs in the proof of the main theorem

We will choose an appropriate δ0 later. We decompose γ̇ into two parts,
Γ1 and Γ2 (cf. Fig. 3), asΓ1 = {µ ∈ γ̇ | |arg µ| < δ0 + π/2},Γ2 = {µ ∈ γ̇ | |arg µ| ≥ δ0 + π/2},

and we set Sj = {µR(µ; A) | µ ∈ Γj } for j = 1, 2. By Lemma 2.3, S1 is
R-bounded with
(3.7)

R(S1 ) ≤ P1 (R(T0 )),where P1 is a polynomial of degree one, defined by (2.3). Note that the set
Tπ/2 in Lemma 2.3 is R-bounded, and its R-bound is equal to that of T0
here. It remains to show that S2 is R-bounded.
We first prove that there exists η > 0 independent of τ such that
(3.8)

kR(µ; A)k ≤ (1 − 2θ)τ η

14

T. Kemmochi

for µ ∈ Γ2 . Take δ1 ∈ (δ0 , δ) sufficiently close to δ so that2(sin δ − sin δ1 ) = ε/2.

(3.9)

We additionally decompose Γ2 into Γ2,1 and Γ2,2 (cf. Fig. 3), whereΓ2,1 = {µ ∈ Γ2 | |arg µ| < δ1 + π/2},

It is well-known thatkR(µ; A)k ≤1,
dist(µ; S(A))Γ2,2 = Γ2 \ Γ2,1 .

µ ∈ C \ S(A),where S(A) is the closure in C (cf. [22, Theorem 3.9 in Chapter 1]). To
compute dist(µ; S(A)), we set r = 1/[(1 − 2θ)τ ], which is the radius of
the circle γ. Assume that µ ∈ Γ2,1 . Then, since µ ∈ Σδ+π/2 and S(A) ⊂C \ Σδ+π/2 , by (NR)δ,ε we havedist(µ; S(A)) ≥ dist(µ; ∂Σδ+π/2 ) = |µ| sin(δ + π/2 − |arg µ|).

Noting that |µ| = 2r sin(|arg µ| − π/2), we have|µ| sin(δ + π/2 − |arg µ|) = 2r sin(|arg µ| − π/2) sin(δ + π/2 − |arg µ|)≥ 2r sin δ0 sin(δ − δ1 ).

Therefore,1(1 − 2θ)τ
=,2r sin δ0 sin(δ − δ1 )2 sin δ0 sin(δ − δ1 )Next, we assume that µ ∈ Γ2,2 . In this case,(3.10) kR(µ; A)k ≤µ ∈ Γ2,1 .

dist(µ; S(A)) ≥ |µ| − r(A) ≥ 2r sin δ1 − r(A).

By the condition (NR)δ,ε and (3.9),

2r sin δ1 − r(A) = [2r sin δ − r(A)] − 2r(sin δ − sin δ1 ) ≥ εr − εr/2 = εr/2,

which implies(1 − 2θ)τ
, µ ∈ Γ2,2 .
ε/2From (3.10) and (3.11), we obtain (3.8) with
12
η = max,.
2 sin δ0 sin(δ − δ1 ) ε
We are now ready to demonstrate the R-boundedness of S2 . Fix µ0 ∈ Γ2 .
Then R(µ0 ; A) can be expanded in a Taylor series as∞
X
R(µ; A) =(µ0 − µ)n R(µ0 ; A)n+1 ,(3.11)

kR(µ; A)k ≤n=0

provided that µ ∈ ρ(A) and |µ − µ0 | < kR(µ0 ; A)k−1 . Setr1
r0 == , S(µ) = {R(ζ; A) | ζ ∈ Γ2 , |ζ − µ| < r0 /4},
(1 − 2θ)τ ηηDiscrete maximal regularity

15

for µ ∈ Γ2 . Then, noting that r0 ≤ kR(µ0 ; A)k−1 by (3.8), we have
∞
o
n X
(µ0 − µ)n R(µ0 ; A)n+1 µ ∈ Γ2 , |µ − µ0 | < r0 /4
R(S(µ0 )) = R
≤
≤

∞
X

n=0

R({(µ0 − µ)n R(µ0 ; A)n+1 | µ ∈ Γ2 , |µ − µ0 | < r0 /4})

n=0
∞ 
X
n=0

1
r0
2

n

n+1

kR(µ0 ; A)k≤∞  nX1
n=02

kR(µ0 ; A)k ≤2.
r0

Here, we have applied [27, Lemma 2.4] in the second step. That is, S(µ) is
R-bounded, and(3.12)

R(S(µ)) ≤ 2(1 − 2θ)τ η

for every µ ∈ Γ2 . Now, we set B(µ) = {ζ ∈ γ | |ζ − µ| < r0 /4} for µ ∈ γ, so
[
that
γ=
B(µ).
µ∈γ

Since γ is compact, there exist N0 ∈ N and µ0 , . . . , µN0 ∈ γ satisfying
γ=

N0
[

B(µj ).

j=0

Moreover, since the ratio of the radii of γ and B(µ) is independent of τ , we
can take N0 independent of τ . Thus,[{R(µ; A) | µ ∈ Γ2 } =S(µj ),
0≤j≤N0µj ∈Γ2

which implies that {R(µ; A) | µ ∈ Γ2 } is R-bounded, andX
R({R(µ; A) | µ ∈ Γ2 }) ≤R(S(µj )) ≤ 2(N0 + 1)(1 − 2θ)τ η
0≤j≤N0µj ∈Γ2by (3.12). Noting that |µ| ≤ 2/[(1 − 2θ)τ ] for µ ∈ γ, we can obtain the
R-boundedness of S2 with the uniform bound(3.13)

R(S2 ) ≤ 8(N0 + 1)η,which implies the uniform R-boundedness of T1 .

Step 4. Finally, we show (3.1) for θ ∈ [0, 1/2). We obtain(3.14)

R(T1 ) ≤ R(S1 ) + R(S2 ) ≤ P1 (R(T0 )) + 8(N0 + 1)η

from (3.7) and (3.13), where P1 (X) is defined by (2.3) with α = R(T0 ) tan δ0 .
By the definition of N0 , it can be seen that
2π.
(3.15)
N0 + 1 ≤1/(4η)

16

T. Kemmochi

Thus, we only need to estimate η. By simple computations, one can obtainandδ − δ1 ≥ sin δ − sin δ1 = ε/4sin(δ − δ1 ) ≥ sin

ε
ε
≥
.
4
2π

Therefore,(3.16)πη ≤ maxε21,
π sin δ0


=

π 1
.
ε sin δ0Now, taking
δ0 = minwe obtainδ1,, arctan2
2R(T0 )

δ 1
α = min R(T0 ) tan ,,2 2which yields

P1 (R(T0 )) ≤ 4 1 +1
tan 2δ+ R(T0 ).

Equations (3.14)–(3.16) imply(3.17)

R(T1 ) ≤ P2 (R(T0 )),where P2 is a polynomial of degree two, and depends only on δ and ε. Hence
we can establish (3.1) from (3.4), (3.5), (3.17), and Theorem 2.11.
From Corollary 2.12 and Theorem 3.2, we can deduce the following assertion. This is a version of our main theorem applicable to the finite element
method.
Corollary 3.3. Let X be a UMD space, and X0 ⊂ X a closed subspace.
Suppose that p ∈ (1, ∞) and θ ∈ [0, 1]. Assume that A ∈ L(X0 ) has maximal
Lp -regularity with a constant CA , and satisfies condition (NR)δ,ε for someδ ∈ (0, π/2) and ε > 0. Then A has maximal lp -regularity with a constant
CDMR , which depends only on p, θ, CA , δ, ε, and X, but is independentof τ , X0 and the operator A.
In view of this corollary, we can assume that X is a Lebesgue space and
X0 is a finite element space.
Roughly speaking, Theorem 3.2 says that continuous maximal regularity implies discrete maximal regularity. This is also true in the opposite
direction, which is a generalization of [5, Theorem 1.1] in a sense. To show
this assertion, we apply Blunck’s result. We refer the reader to [5, Proposition 1.4] for the proof. Although Blunck did not mention the dependence of
C below, one can obtain it by tracing the proof carefully.

Discrete maximal regularity

17

Proposition 3.4. Let X be a Banach space and M ∈ L∞ (T; L(X)).
Suppose that the operator TM defined by (2.10) can be extended to a bounded
operator on lp (Z; X) for some p ∈ (1, ∞). Then the set
TM = {M (z) | z is a Lebesgue point of M }is R-bounded with
R(TM ) ≤ CkTM kL(lp (Z;X)) ,where C > 0 depends only on p. Here, we denote the extension of TM by
the same symbol.
Theorem 3.5. Let X be a UMD space, p ∈ (1, ∞) and θ ∈ [0, 1]. Assume
0 , and condition
that A ∈ L(X) has maximal lp -regularity with a constant CAp(S) is fulfilled when 0 ≤ θ < 1/2. Then A has maximal L -regularity with a
0 and X.
constant CMR depending only on p, CA
Proof. Since A has maximal lp -regularity, the operator TMτ defined by(2.10) is bounded in lp (Z; X) uniformly in τ > 0. Combining this with
Proposition 3.4, we obtain
0
R(TMτ ) ≤ cp CA,

where cp > 0 depends only on p and∀τ > 0,TMτ = {Mτ (λ) | λ ∈ T \ {1}}and Mτ is defined by (2.9).
Now, we show that the set T0 = {µR(µ; A) | µ ∈ iR \ {0}} is Rbounded. Recall that (3.3) and (2.14) hold. Therefore, no further argument
is needed when θ = 1/2. Now assume that 1/2 < θ ≤ 1. Set hθ,τ (ζ) =(1 − e(2θ−1)τ z )/[(2θ − 1)τ ] for ζ ∈ C. Then it is easy to see that hθ,τ (µ) ∈gθ,τ (T \ {1}) for µ ∈ iR \ {0} and hθ,τ (µ) → µ as τ ↓ 0. Thus, for n ∈ N,
x0 , . . . , xn ∈ X, and µ0 , . . . , µn ∈ iR \ {0}, we obtain
1

n
X

0

j=0

rj (t)µj R(µj ; A)xj dt

= lim
τ ↓0

1

n
X

0

j=0

0
rj (t)hθ,τ (µj )R(hθ,τ (µj ); A)xj dt ≤ cp CA

1

n
X

0

j=0

rj (t)xj dt,

which implies that T0 is R-bounded. Here, we have applied Lebesgue’s convergence theorem in the first equality. The dependence of CMR on X comesfrom the continuous version of the vector-valued Fourier multiplier theorem [27, Theorem 3.4]. The proof when 0 ≤ θ < 1/2 is almost the same.
Recall that maximal Lp -regularity is independent of p ∈ (1, ∞). That is,
if an operator A has maximal Lp0 -regularity for some p0 ∈ (1, ∞), then it has

18

T. Kemmochi

maximal Lp -regularity for all p ∈ (1, ∞). Combining this with Theorems 3.2
and 3.5, we can derive p-independence for maximal lp -regularity.
Corollary 3.6. Let X be a UMD space, A ∈ L(X) the infinitesimal
generator of a bounded analytic semigroup on X, and θ ∈ [0, 1]. Assume
that condition (NR)δ,ε is fulfilled when 0 ≤ θ < 1/2, and A has maximal
lp0 -regularity for some p0 ∈ (1, ∞). Then A has maximal lp -regularity for
all p ∈ (1, ∞).
4. The backward Euler method. In the case of the backward Euler
method, we can establish several more properties analogous to those investigated in the previous section. Moreover, we can remove a restriction on
operators. In our main theorem we have assumed that the operator A isbounded, since the Cauchy problem (2.6) is ill-posed in general. However,(2.6) is well-posed for unbounded operators when θ = 1, whenever I − τ A
is invertible. Therefore, we need not assume that A is bounded.
This section concerns two independent topics. We first characterize discrete maximal regularity for unbounded operators. The result is, in a sense,
an extension of Blunck’s characterization of discrete maximal regularity for
power-bounded operators, and includes the result of [3]. The next topic
is the derivation of an a priori estimate for non-zero initial values. In thecontinuous case, it is well-known that an a priori estimate (2.5) is valid
for non-zero initial values with some modifications. With this in mind, weestablish an estimate similar to (2.8) with appropriate initial values. We remark that our results on this topic are minor modifications of results in [4,Chapter 2].
4.1. Characterization of discrete maximal regularity. In [5],Blunck considered discrete maximal regularity for the forward Euler method,and characterized it as continuous maximal regularity. However, his proof
is valid only in the case where the operator A is bounded, as long as theforward Euler method is considered.
Our aim is to characterize discrete maximal regularity for unbounded
operators. Therefore, we consider the backward Euler method: un+1 − un
= Aun+1 + f n+1 , n ∈ N,τ 0
u = 0,
so that the iteration operator Tτ in (2.11) is just the a resolvent of A, i.e.,Tτ = (I − τ A)−1 = τ −1 R(τ −1 ; A)

if the operator I − τ A is invertible. This scheme was considered by Ashyralyev et al. [3] and it was shown that continuous maximal regularity implies
discrete maximal regularity. However, the converse was not considered.

Discrete maximal regularity

19

Now, we are in a position to characterize discrete maximal regularity insimilar terms to Blunck. The following theorem corresponds to Blunck’s [5,Theorem 1.1]. Note that I − τ A is invertible for each τ > 0 if A generates abounded semigroup.
Theorem 4.1. Let X be a UMD space, and let A be a linear operator
on X that generates a bounded analytic semigroup T (t) on X. Set Tτ =(I − τ A)−1 for τ > 0. Then the following statements are equivalent:(a) A has discrete maximal regularity for θ = 1.
(b) {(λ − 1)Tτ R(λ; Tτ ) | λ ∈ T \ {1}} is R-bounded uniformly for τ > 0.
(c) {Tτn , n(Tτ − I)Tτn | n ∈ N} is R-bounded uniformly for τ > 0.
(d) A has continuous maximal regularity.
(e) {µR(µ; A) | µ ∈ iR \ {0}} is R-bounded.
(f) {T (t), tAT (t) | t > 0} is R-bounded.

Proof. The equivalences (d)⇔(e)⇔(f) are due to Weis (Theorem 2.5).
The implications (e)⇒(b)⇒(a) are due to Ashyralyev et al. [3], and (a)⇒(b)⇒(e) are shown in Theorem 3.5. Hence, it suffices to show that (f)⇒(c) and(c)⇒(f).
(f)⇒(c). It suffices to show that the set {Tτn , n(Tτ − I)Tτn | n ∈ N,n ≥ 1} is R-bounded uniformly in τ > 0. It is well-known that∞1nR(λ; A) =tn−1 e−λt T (t) dt(n − 1)!
0in L(X) for λ ∈ ρ(A) with Re λ > 0 and n ≥ 1. Therefore,∞1
tn−1 e−t/τ T (t) dt
Tτn = nτ (n − 1)!
0and, noting that Tτ − I = τ ATτ ,n(Tτ − I)Tτn =
1tn−1 e−t/τ tAT (t) dtnτ (n − 1)!
∞0∞

for τ > 0 and n ≥ 1. These equations imply (c), by the formula 0 tn e−αt dt= n!α−n−1 for α > 0 and n ∈ N, and by the convexity property ofR-boundedness [19, Theorem 2.13].
(c)⇒(f). Set Sτ = {Tτn , n(Tτ − I)Tτn | n ∈ N}, and assume that there
exists C > 0 independent of τ > 0 such that R(Sτ ) ≤ C for each τ . DefineAτ = (Tτ − I)/τ , which is the Yosida approximation of A. Then, as is
well-known,(4.1)
lim etAτ x = T (t)x in Xτ ↓0for every t > 0 and every x ∈ X. Moreover, the convergenceis uniform onPeach bounded interval. Here, for B ∈ L(X), eB = n∈N B n /n! is the usual
exponential of B. Now, we show that the set {T (t)}t is R-bounded. Since

20

T. Kemmochi

e

tAτ

−t/τ

=e

 
∞
X
1 t n
n=0

n! τ

Tτn ∈ CH(Sτ ),the set {etAτ | t > 0} is R-bounded, and its R-bound does not exceed C,
where CH(·) is the convex hull and the overline indicates closure in the strong
topology of L(X). Thus, for N ∈ N, sj > 0, and xj ∈ X (j = 0, . . . , N ), we
have
1 X
1 X
1 X
N
N
N
rj (t)T (sj )xj dt = limrj (t)esj Aτ xj dt ≤ C
rj (t)xj dt,
0

τ ↓0

j=0

0

j=0

0

j=0

which implies that {T (t)}t is R-bounded.
We next establish the R-boundedness of {tAT (t)}t . Note that for everyt > 0 and x ∈ X, one can obtain
lim tAτ etAτ x = tAT (t)x in Xτ ↓0

in a way similar to the proof of (4.1), and the convergence is uniform on eachbounded interval. We claim that the set Sτ0 = {(n + 1)(Tτ − I)Tτn | n ∈ N}
is R-bounded. Indeed, sinceSτ0 = {Tτ − I} ∪ {(1 + n−1 )nTτn | n ∈ N, n ≥ 1},
the set Sτ0 is R-bounded withR(Sτ0 ) ≤ kTτ k + 1 + 2R(Sτ ) ≤ 1 + 3C.
Therefore, since ∞X
1 t n+1
tAτ
−t/τ
tAτ e
=e
(Tτ − I)Tτnn! τn=0 n+1∞Xt
1−t/τ
(n + 1)(Tτ − I)Tτn ∈ CH(Sτ0 ),=e(n + 1)! τ

n=0
tA
τ
the set {tAτ e }t is R-bounded, and its R-bound does not exceed 1 + 3C.

This implies the R-boundedness of {tAT (t)}t in the same way as above.

4.2. A priori estimate with non-zero initial values. Let X be a
Banach space, and let A be a linear operator on X. In the theory of nonlinear
evolution equations, the choice of initial value is important. Therefore, weneed to obtain an a priori estimate of maximal regularity (2.5) with nonzero initial values. It is known that the desired estimate is valid for u(0) ∈(X, D(A))1−1/p,p , which is the real interpolation space provided 0 ∈ ρ(A).
The estimate is as follows:(4.2)
ku0 kLp (R+ ;X) + kAukLp (R+ ;X) ≤ C(kf kLp (R+ ;X) + ku0 k1−1/p,p ),where u is the solution ofu0 (t) = Au(t) + f (t), t > 0,u(0) = u0 ,Discrete maximal regularity

21

with f ∈ Lp (R+ ; X) and u0 ∈ (X, D(A))1−1/p,p . Here, k·k1−1/p,p is the usual
norm of (X, D(A))1−1/p,p .
We present the discrete version of (4.2) in the case of the backward Euler
method. This problem in a bounded interval has already been considered by
Ashyralyev and Sobolevskiı̆ [4, Chapter 2]. We obtain the same estimate in
the case where the interval is unbounded. We consider the following problem:
( n+1
u
− un
= Aun+1 + f n+1 , n ∈ N,
(4.3)
τ
u0 = u0
for f = (f n ) ∈ lp (N; X) and u0 ∈ (X, D(A))1−1/p,p , with p ∈ (1, ∞). Recall
that v1 = (v n+1 )n for v = (v n ) ∈ X N .
Theorem 4.2. Let X be a Banach space, and let A be a linear operator
on X. Suppose that p ∈ (1, ∞) and τ > 0. Assume that 0 ∈ ρ(A) and A has
discrete maximal regularity for θ = 1. Then, for each f = (f n ) ∈ lp (N; X)
and u0 ∈ (X, D(A))1−1/p,p , there exists a unique solution u = (un ) of (4.3)
satisfyingkDτ uklτp (N;X) + kAu1 klτp (N;X) ≤ C(kf1 klτp (N;X) + ku0 k1−1/p,p ),where C > 0 is independent of τ , f , and u0 .
This theorem can be deduced from the following embedding result. The
proof is essentially the same as in [4, Theorem 3.1 in Chapter 2]. Thus weomit it. Recall that Tτ = (I − τ A)−1 .
Lemma 4.3. Let X be a Banach space, and let A be a linear operator
on X. Suppose that p ∈ (1, ∞) and τ > 0. Assume that 0 ∈ ρ(A) and A
generates a bounded analytic semigroup on X. Define a Banach space Eτp as∞noXEτp = x ∈ X
kATτn+1 xkpX < ∞n=0with the norm
kxkEτp = kxkX +

∞
X
n=0

kATτn+1 xkpX

1/p

.

Then the embedding(X, D(A))1−1/p,p ,→ Eτpholds uniformly for τ > 0 and X.
Proof of Theorem 4.2. We can establish the desired estimate by dividing
the problem (4.3) into two problems(Dτ v)n = Av n+1 + f n+1 , n ∈ N,
(Dτ w)n = Awn+1 , n ∈ N,andv 0 = 0,w0 = u0 ,and utilizing Theorem 3.2 and Lemma 4.3.

22

T. Kemmochi

We conclude this paper by stating an applicable version of Theorem 4.2.
The corollary below can be established in the same way as Corollary 3.3
since the embedding constant of Lemma 4.3 is independent of X.
Corollary 4.4. Let X be a Banach space, X0 ⊂ X a closed subspace,and A a linear operator on X0 . Suppose that p ∈ (1, ∞) and τ > 0. Assume
that 0 ∈ ρ(A) and A has discrete maximal regularity for θ = 1. Then, foreach f = (f n ) ∈ lp (N; X0 ) and u0 ∈ (X0 , D(A))1−1/p,p , there exists a uniquesolution u = (un ) of (4.3) satisfyingkDτ uklτp (N;X) + kAu1 klτp (N;X) ≤ C(kf1 klτp (N;X) + ku0 k1−1/p,p ),where C > 0 is independent of τ , f , u0 , and the Banach space X0 . Here,the norm of u0 is that of (X, D(A))1−1/p,p .
Acknowledgements. I would like to thank Professor Norikazu Saito
for bringing this topic to my attention, suggesting the modification of theproof of the main theorem, and encouraging me through valuable discussions. I also thank the anonymous reviewer for valuable comments and suggestions to improve the quality of the paper.
This work was supported by JSPS KAKENHI Grant Number 15J07471,
and the Program for Leading Graduate Schools, MEXT, Japan.

References
[1]
[2]
[3]

[4]
[5]
[6]
[7]
[8]

[9]

R. P. Agarwal, C. Cuevas, and C. Lizama, Regularity of Difference Equations on
Banach Spaces, Springer, 2014.
H. Amann, Quasilinear parabolic problems via maximal regularity, Adv. Differential
Equations 10 (2005), 1081–1110.
A. Ashyralyev, S. Piskarev, and L. Weis, On well-posedness of difference schemes
for abstract parabolic equations in Lp ([0, T ]; E) spaces, Numer. Funct. Anal. Optim.
23 (2002), 669–693.
A. Ashyralyev and P. E. Sobolevskiı̆, Well-Posedness of Parabolic Difference Equations, Birkhäuser, Basel, 1994.
S. Blunck, Maximal regularity of discrete and continuous time evolution equations,
Studia Math. 146 (2001), 157–176.
J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat. 21 (1983), 163–168.
D. L. Burkholder, A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional, Ann. Probab. 9 (1981), 997–1011.
D. L. Burkholder, A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions, in: Conference on Harmonic Analysis
in Honor of Antoni Zygmund, Vol. I (Chicago, IL, 1981), Wadsworth, Belmont, CA,
1983, 270–286.
C. Cuevas and C. Lizama, Semilinear evolution equations on discrete time andmaximal regularity, J. Math. Anal. Appl. 361 (2010), 234–245.

Discrete maximal regularity
[10]

[11]
[12]

[13]
[14]

[15]
[16]
[17]
[18]
[19]

[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]

23

G. Dore, Lp regularity for abstract differential equations, in: Functional Analysis
and Related Topics (Kyoto, 1991), Lecture Notes in Math. 1540, Springer, Berlin,
1993, 25–38.
G. Dore and A. Venni, On the closedness of the sum of two closed operators, Math.
Z. 196 (1987), 189–201.
K. Eriksson, C. Johnson, and S. Larsson, Adaptive finite element methods for parabolic problems. VI. Analytic semigroups, SIAM J. Numer. Anal. 35 (1998), 1315–
1325.
H. Fujita, N. Saito, and T. Suzuki, Operator Theory and Numerical Methods, NorthHolland, Amsterdam, 2001.
I. P. Gavrilyuk and V. L. Makarov, Exponentially convergent algorithms for theoperator exponential with applications to inhomogeneous problems in Banach spaces,
SIAM J. Numer. Anal. 43 (2005), 2144–2171.
M. Geissert, Discrete maximal Lp regularity for finite element operators, SIAM J.
Numer. Anal. 44 (2006), 677–698.
M. Geissert, Applications of discrete maximal Lp regularity for finite element operators, Numer. Math. 108 (2007), 121–149.
T. Kemmochi and N. Saito, Discrete maximal regularity and the finite element
method for parabolic equations, arXiv:1602.06864 (2016).
B. Kovács, B. Li, and C. Lubich, A-stable time discretizations preserve maximal
parabolic regularity, arXiv:1511.07823 (2015).
P. C. Kunstmann and L. Weis, Maximal Lp -regularity for parabolic equations, Fourier
multiplier theorems and H ∞ -functional calculus, in: Functional Analytic Methods for
Evolution Equations, Lecture Notes in Math. 1855, Springer, Berlin, 2004, 65–311.
B. Li, Maximum-norm stability and maximal Lp regularity of FEMs for parabolic
equations with Lipschitz continuous coefficients, Numer. Math. 131 (2015), 489–516.
Yu. I. Mokin, A discrete analogue of the multiplier theorem, Zh. Vychisl. Mat. i Mat.
Fiz. 11 (1971), 746–749 (in Russian).
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential
Equations, Springer, New York, 1983.
P. Portal, Maximal regularity of evolution equations on discrete time scales, J. Math.
Anal. Appl. 304 (2005), 1–12.
N. Saito, Conservative upwind finite-element method for a simplified Keller–Segelsystem modelling chemotaxis, IMA J. Numer. Anal. 27 (2007), 332–365.
N. Saito, Error analysis of a conservative finite-element approximation for the
Keller–Segel system of chemotaxis, Comm. Pure Appl. Anal. 11 (2012), 339–364.
Y. Shibata and S. Shimizu, Lp -Lq maximal regularity and viscous incompressible
flows with free surface, Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), 151–155.
L. Weis, Operator-valued Fourier multiplier theorems and maximal Lp -regularity,
Math. Ann. 319 (2001), 735–758.

Tomoya Kemmochi
Graduate School of Mathematical Sciences
The University of Tokyo
Komaba 3-8-1, Meguro-ku
Tokyo, 153-8914, Japan
E-mail: kemmochi@ms.u-tokyo.ac.jp
URL: http://t-kemmochi.github.io/