Math. Z. 240, 311–343 (2002)
Digital Object Identifier (DOI) 10.1007/s002090100384
The operator-valued Marcinkiewicz multiplier
theorem and maximal regularity
Wolfgang Arendt1 , Shangquan Bu1,2
1
2
Abteilung Angewandte Analysis, Universität Ulm, 89069 Ulm, Germany
(e-mail: [email protected])
Department of Mathematical Science, University of Tsinghua, 100084 Beijing, China
(e-mail: [email protected])
Received: 21 December 2000; in final form: 12 June 2001 /
c Springer-Verlag 2002
Published online: 1 February 2002 – Abstract. Given a closed linear operator on a U M D-space, we characterize
maximal regularity of the non-homogeneous problem
u + Au = f
with periodic boundary conditions in terms of R-boundedness of the resolvent. Here A is not necessarily generator of a C0 -semigroup. As main tool
we prove an operator-valued discrete multiplier theorem. We also characterize maximal regularity of the second order problem for periodic, Dirichlet
and Neumann boundary conditions.
Classical theorems on Lp -multipliers are no longer valid for operator-valued
functions unless the underlying space is isomorphic to a Hilbert space
(see Sect. 1 for precise statements of this result). However, recent work
of Clément-de Pagter-Sukochev-Witvliet [CPSW], Weis [W1], [W2] and
Clément-Prüss [CP] show that the right notion in this context is R-boundedness of sets of operators. This condition is strictly stronger than boundedness
in operator norm (besides in the Hilbert space) and may be defined with help
of the Rademacher functions. And indeed, Weis [W1] showed that Mikhlin’s
classical theorem on Fourier multipliers on Lp (R; X) holds if boundedness
is replaced by R-boundedness (see [CP] for another proof based on results
of Clément-de Pagter-Sukochev and Witvliet [CPSW]).
This research is part of the DFG-project: “Regularität und Asymptotik für elliptische und
parabolische Probleme”. The second author is supported by the Alexander-von-Humboldt
Foundation and the NSF of China.
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W. Arendt, S. Bu
Motivation of these investigations are regularity problems for differential
equations in Banach spaces. Given the generator A of a holomorphic C0 semigroup, the problem of maximal regularity of the inhomogeneous problem
u (t) = Au(t) + f (t) t ∈ [0, 1]
P0
u(0) = 0
with Dirichlet boundary conditions obtained much attention since the pioneering articles of Da Prato-Grisvard [DG] and Dore-Venni [DV]. And
indeed, it is now possible to characterize maximal regularity of the problem
P0 in terms of R-boundedness of the resolvent (see Weis [W1], [W2] and
Clément-Prüss [CP]). In the present article we study maximal regularity of
the inhomogeneous problem with periodic boundary conditions
u (t) = Au(t) + f (t) t ∈ [0, 2π]
Pper
u(0) = u(2π) .
Now it is no longer natural to suppose that A is the generator of a C0 semigroup. We merely assume that A is a closed operator on a U M D-space.
One of our main results (Theorem 2.3) says that Pper is strongly Lp -wellposed for 1 < p < ∞ if and only if the set {k(ik − A)−1 : k ∈ Z} is
R-bounded.
In order to treat the periodic case we need a multiplier theorem in the discrete case. Our main result of Sect. 1 is an operator-valued version of the
Marcinkiewicz theorem which is very easy to formulate. It turns out that this
discrete multiplier theorem is not only suitable for the treatment of the periodic problem Pper but gives an alternative approach to maximal regularity
for P0 (Sect. 5). It is possible to deduce our discrete multiplier theorem from
a more complicated version by S̆traklj and Weis [SW] whose formulation
and proof are quite involved. So we prefer to give a direct and easy proof in
Sect. 1.
Even though it became clear now that R-boundedness of resolvents is the
right notion for maximal regularity, it is not easy to verify this condition in
concrete cases. In Sect. 4 we show how R-boundedness of |s|θ (is−A)−1 for
θ ∈ (0, 1) can be deduced from boundedness of |s|(is − A)−1 (s ∈ R).
This is used to prove that the mild solutions of Pper are Hölder continuous. Again this result is true for arbitrary closed operators. We need some
preparation to clarify the notion of mild solution in Sect. 3. If A generates
a C0 -semigroup T , then it can be defined with help of the variation of constant formula, and by a result of Prüss [Pr], mild well-posedness of Pper is
equivalent to (I − T (2π)) being invertible. We show in Sect. 3 that this in
turn is equivalent to ((ik − A)−1 )k∈Z being an Lp -multiplier. An analogous
continuous version of this is proved by Latushkin and Shvydkoy [LS] in
recent work.
The operator-valued Marcinkiewicz multipler theorem
313
Finally, in Sect. 6 we characterize strong Lp -well-posedness of the second order Cauchy problem with periodic, Dirichlet and Neumann boundary
conditions in terms of R-boundedness. Again the results are valid for arbitrary closed operators.
Acknowledgements. The authors thank the referee for several valuable suggestions and
comments. They are most grateful to C. Le Merdy for illuminating information on Pisier’s
inequality and lacunary multipliers (cf. end of section 1).
1. The operator-valued Marcinkiewicz multiplier theorem
Let X be a complex Banach space. We consider the Banach space
Lp (0, 2π; X) with norm
 2π
 p1
f p :=  f (t)p dt
0
where 1 ≤ p < ∞. For f ∈
Lp (0, 2π; X)
1
fˆ(k) :=
2π
2π
we denote by
e−ikt f (t)dt
0
the k-th Fourier coefficient of f , where k ∈ Z. For k ∈ Z, x ∈ X we
let ek (t) = eikt and (ek ⊗ x)(t) = ek (t)x (t ∈ R). Then for xk ∈ X,
k = −m, −m + 1, . . . , m,
f=
m
ek ⊗ xk
k=−m
is the trigonometric polynomial given by
f (t) =
m
eikt xk
(t ∈ R) .
k=−m
Then fˆ(k) = 0 if |k| > m. The space T(X) of all trigonometric polynomials is dense in Lp (0, 2π; X). In fact, let f ∈ Lp (0, 2π; X). Then by Fejer’s
theorem, one has
f = lim σn (f )
(1.1)
n→∞
in
Lp (0, 2π; X)
where
n
m
1 σn (f ) :=
ek ⊗ fˆ(k) .
n+1
m=0 k=−m
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W. Arendt, S. Bu
As an immediate consequence we note the Uniqueness Theorem. Let f ∈
L1 (0, 2π; X).
a) If fˆ(k) = 0 for all k ∈ Z, then f (t) = 0 t−a.e.
b) If fˆ(k) = 0 for all k ∈ Z \ {0}, then f (t) = fˆ(0) t−a.e.
Let X, Y be Banach spaces and let L(X, Y ) be the set of all bounded linear
operators from X to Y . If (Mk )k∈Z ⊂ L(X, Y ) is a sequence, we consider
the associated linear mapping
M : T(X) → T(Y )
given by
M
=
ek ⊗ xk
k
ek ⊗ Mk xk .
k
We say that the sequence (Mk )k∈Z is an Lp -multiplier, if there exists a
constant C such that
ek ⊗ Mk xk ≤ C ek ⊗ xk k
k
p
for all trigonometric polynomials
k
p
ek ⊗ xk .
Proposition 1.1. Let (Mk )k∈Z ⊂ L(X, Y ) be a sequence, then the following two statements are equivalent
(i) (Mk )k∈Z is an Lp -multiplier.
(ii) For each f ∈ Lp (0, 2π; X) there exists g ∈ Lp (0, 2π; Y ) such that
ĝ(k) = Mk fˆ(k) for all k ∈ Z .
In that case there exists a unique operator M ∈ L(Lp (0, 2π; X),
Lp (0, 2π; Y )) such that
(1.2)
(M f )ˆ(k) = Mk fˆ(k)
(k ∈ Z)
for all f ∈ Lp (0, 2π; X). We call M the operator associated with (Mk )k∈Z .
One has
n
m
1 M f = lim
ek ⊗ Mk fˆ(k)
(1.3)
n→∞ n + 1
m=0 k=−m
in Lp (0, 2π; Y ) for all f ∈ Lp (0, 2π; X).
The operator-valued Marcinkiewicz multipler theorem
315
Proof.
(i) ⇒ (ii). Define M̃ : T(X) → T(Y ) by M̃ ( ek ⊗ xk ) =
ek ⊗ Mk xk . Then by the assumption, M̃ has a unique extension M ∈
L(X, Y ). Then (1.3) follows by continuity. Clearly, (1.3) implies (1.2).
(ii) ⇒ (i). Define M f = g with ĝ(k) = Mk fˆ(k) (k ∈ Z). Then the
uniqueness theorem and closed graph theorem show that
M ∈ L(Lp (0, 2π; X), Lp (0, 2π; Y )).
Let 1 ≤ q < ∞. Denote by rj the j-th Rademacher function on [0, 1].
For x ∈ X we denote by rj ⊗ x the vector-valued function t → rj (t)x.
Definition 1.2. A family T ⊂ L(X, Y ) is called R-bounded if there exist
cq ≥ 0 such that
n
n
(1.4)
rj ⊗ Tj xj ≤ cq rj ⊗ xj j=1
j=1
Lq (0,1;X)
Lq (0,1;X)
for all T1 , . . . , Tn ∈ T, x1 , · · · , xn ∈ X and n ∈ N, where 1 ≤ q < ∞.
By Kahane’s inequality [LT, Theorem 1.e.13] if such constant cq exists for
some q ∈ [1, ∞), there also exists such constant for all q ∈ [1, ∞). We
denote by Rq (T) the smallest constant cq such that (1.4) holds. Sometimes
we say that T is R-bounded in L(X, Y ) to be more precise.
The concept of R-boundedness (read Rademacher boundedness or randomized boundedness) was introduced by Bourgain [Bo]. It is fundamental to recent work of Clément-de Pagter-Sukochev-Witvliet [CPSW], Weis
[W1], [W2], S̆trkalj-Weis [SW] and Clément-Prüss [CP]. We will use several
basic results of [CPSW].
Now we can formulate the following multiplier theorem which is the
discrete analog of the operator-valued version of Mikhlin’s theorem due to
Weis [W1] (see also [CP]).
Theorem 1.3 (Marcinkiewicz operator-valued multiplier theorem). Let
X, Y be U M D-spaces. Let Mk ∈ L(X, Y ) (k ∈ Z). If the sets {k(Mk+1 −
Mk ) : k ∈ Z} and {Mk : k ∈ Z} are R-bounded, then (Mk )k∈Z is an Lp multiplier for 1 < p < ∞.
We need the following definition. Let N0 = N ∪ {0}.
Definition 1.4. An unconditional Schauder composition of X is a family
{∆k : k ∈ N0 } of projection in L(X) such that
(a) ∆k ∆ = 0 if k = &
∞
(b)
∆π(k) x = x for all x ∈ X and for each permutation π : N0 → N0 .
k=0
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W. Arendt, S. Bu
The basic example for our purposes is the following
Example 1.5. Let X be a U M D-space. For k ∈ N define ∆k f =
em ⊗ fˆ(m) and ∆0 f = e−1 ⊗ f(−1) + e0 ⊗ f(0) + e1 ⊗ f(1),
2k ≤|m|<2k+1
(f ∈ Lp (0, 2π; X)). Then (∆k )k∈N0 is an unconditional Schauder decomposition of Lp (0, 2π; X).
The proof of this result is due to Bourgain [Bo]. However, in the case
where X = Lp (Ω), 1 < p < ∞, it can be deduced from the scalar case
which is a central part of classical Littlewood-Paley theory (see [EG]).
The following lemma due to Clément-de Pagter-Sukochev-Witvliet
[CPSW, Theorem 3.4] gives a sufficient condition for multipliers with respect to an unconditional Schauder decomposition.
Proposition 1.6. Let (∆k )k∈N0 be an unconditional Schauder decomposition of a Banach space X. Let {Tk : k ∈ N0 } ⊂ L(X) be an R-bounded
sequence such that
Tk ∆k = ∆k Tk
for all k ∈ N. Then
Tx =
∞
Tk ∆k x
k=0
converges for all x ∈ X and defines an operator T ∈ L(X).
Besides Proposition 1.6 we need the following properties for the proof of
the multiplier theorem.
Lemma 1.7 (Kahane’s contraction principle [LT]). One has
m
m
rj ⊗ λj xj ≤ 2 max |λj | rj ⊗ xj j=1,...,m
j=1
j=1
p
p
for all λ1 , . . . , λm ∈ C, x1 , . . . , xm ∈ X.
Lemma 1.8. Let S, T ⊂ L(X) be R-bounded sets. Then S · T = {S · T :
S ∈ S, T ∈ T} is R-bounded and
Rp (ST ) ≤ Rp (S) · Rp (T) .
This is easy to see.
Lemma 1.9 ([CPSW, Lemma 3.2]). If S ⊂ L(X, Y ) is R-bounded, then
Rp (coS) ≤ 2Rp (S)
The operator-valued Marcinkiewicz multipler theorem
317
(1 ≤ p < ∞) where


m
m


λ j S j : S j ∈ S , λj ∈ C ,
|λj | ≤ 1, m ∈ N .
coS =


j=1
j=1
If X is a U M D-space, then by a result of Burkholder [Bur],
N
N
ek ⊗ xk =
ek ⊗ xk
R
k=−N
k=0
defines an Lp -multiplier R for 1 < p < ∞, which is called the Rieszprojection.
We need the following easy consequence.
Lemma 1.10. Let X be a U M D-space and 1 < p < ∞. Define the projections P on Lp (0, 2π; X) by
P (
ek ⊗ xk ) =
ek ⊗ xk .
k∈Z
k≥
Then the set {P : & ∈ Z} is R-bounded in L(Lp (0, 2π; X)).
Proof. For n ∈ Z let Sn ∈ L(Lp (0, 2π; X)) be Sn f = e−n · f . Then
P = S− RS . Since R is a bounded operator, it suffices to show that the set
{S : & ∈ Z} is R-bounded in L(Lp (0, 2π; X)). This is an easy consequence
of Kahane’s contraction principle.
Proof of Theorem 1.3. a) We assume that X = Y . Let Z = {f ∈
Lp (0, 2π; X) : fˆ(k) = 0 for all k < 0}. Since the Riesz-projection is
bounded it suffices to show that for some constant C > 0
N
N
ek ⊗ Mk xk ≤ C ek ⊗ xk k=0
p
k=0
p
whenever x0 , . . . , xN ∈ X. Define Qn ∈ L(Z) by
Qn f =
ek ⊗ fˆ(k) for n ∈ N
2n−1 ≤k<2n
and Q0 f = e0 ⊗ fˆ(0). It follows from Example 1.5 and the boundedness
of the Riesz-projection, that the sequence (Qn )n∈N0 is an unconditional
Schauder decomposition of Z. For each k ∈ N0 define Ak ∈ L(Z) by
(Ak f )(t) = Mk f (t) (t ∈ [0, 2π]). It follows from the assumption and
Fubini’s Theorem that the sets {Ak : k ∈ Z} and {k(Ak+1 − Ak ) : k ∈ Z}
are R-bounded in L(Z).
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W. Arendt, S. Bu
Now let f =
Tf =
k≥0
ek ⊗ xk ∈ Z be a trigonometric polynomial. Let
ek ⊗ Mk xk
k≥0
=


Qn 
ek ⊗ Mk xk 
n≥0

=
k≥0
n≥1
n −1
2

(Pk − Pk+1 )Ak · Qn 
k=2n−1
+(P0 − P1 )Q0
ek ⊗ xk
k≥0
ek ⊗ xk .
k≥0
Thus
T =
n≥1
=
n −1
2
Ak (Pk − Pk+1 ) Qn + (P0 − P1 )A0 Q0
k=2n−1
A2n−1 P2n−1 Qn −
n≥1
+
n≥1


n≥1
n −1
2
A2n −1 P2n Qn

(Ak − Ak−1 )Pk  Qn + (P0 − P1 )A0 Q0
k=2n−1 +1
Since (Qn )n∈N0 is an unconditional
Schauder decomposition
of Z by Propo
sition 1.6 and Lemma 1.8,
A2n−1 P2n−1 Qn ,
A2n −1 P2n Qn and (P0 −
n≥1
n≥1
P1 )A0 Q0 define bounded linear operators on Z. In order to estimate the third
2n
−1
1
term observe that
k−1 ≤ 1. Hence by Lemma 1.9 and 1.8,
k=2n−1 +1


n −1
 2

Pk (Ak − Ak−1 ) : n ∈ N 
Rp 
 n−1

k=2
+1


n −1
 2

1

(k − 1)(Ak − Ak−1 )Pk : n ∈ N 
= Rp
 n−1 k − 1

k=2
+1
≤ 2Rp ({(k − 1)(Ak − Ak−1 )Pk : k ∈ N}
≤ 2Rp {k(Ak+1 − Ak ) : k ∈ Z} · Rp ({Pk : k ∈ Z}) < ∞ .
This finishes the proof if X = Y .
b) Now we consider the general case. Since X and Y are U M D-spaces, also
The operator-valued Marcinkiewicz multipler theorem
319
X ⊕Y is a U M D-space. Define M̃k ∈ L(X ⊕Y ) by M̃k (x, y) = (0, Mk x).
It follows from the case a) that (M̃k )k∈Z is an Lp -multiplier. It is easy to see
that this implies that (Mk )k∈Z is an Lp -multiplier (1 < p < ∞).
Next we show that, conversely, R-boundedness is a necessary condition
for Lp -multipliers.
Proposition 1.11. Let X be a Banach space and let (Mk )k∈Z be an Lp multiplier, where 1 ≤ p < ∞. Then the set {Mk : k ∈ Z} is R-bounded.
Proof. By Kahane’s contraction principle we have for ηj ∈ R, xj ∈ X
iη
rj ⊗ e j xj ≤ 2
rj ⊗ xj p
p
j
j
L (0,1;X)
L (0,1;X)
−iηj iηj
= 2
rj ⊗ e
e xj p
j
L (0,1;X)
≤ 4
rj ⊗ eiηj xj .
p
j
L (0,1;X)
By assumption there exists c ≥ 0 such that
ek ⊗ Mk xk ≤ c
ek ⊗ xk k
Lp (0,2π;X)
Hence for all t ∈ [0, 2π]
rj ⊗ Mj xj p
j
L (0,1;X)
k
.
Lp (0,2π;X)
≤ 2
rj ⊗ ej (t)Mj xj j
Integrating over t ∈ [0, 2π] yields
p
2π rj ⊗ Mj xj ≤
j
p
L (0,1;X)
p
2π 1 p
2
rj (s)ej (t)Mj xj dsdt =
0
0 j
p
1 2π p
ej (t)rj (s)Mj xj 2
dtds ≤
0
0
j
Lp (0,1;X)
.
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W. Arendt, S. Bu
p
p p
2 c
ej (t)rj (s)xj dtds =
0
j
p
2π p p
rj ⊗ ej (t)xj dt ≤
2 c
0
j
p
L (0,1;X)
p
2π p p p
2 c 2
rj ⊗ xj dt =
0
j
p
L (0,1;X)
p p p
2 c 2 2π
rj ⊗ xj pLp (0,1;X) .
1 j
Thus Rp {Mk : k ∈ Z} ≤ 4c.
We conclude this section by several comments about optimality of the
operator-valued Marcinkiewicz theorem above (Theorem 1.3).
First of all we remark that on a Hilbert space X each bounded sequence
(Mk )k∈Z ⊂ L(X) is an L2 -multiplier. This follows from the fact that the
Fourier transform given by
f ∈ L2 (0, 2π; X) → (fˆ(k))k∈Z ∈ &2 (X)
is an isometric isomorphism if X is a Hilbert space. On the other hand, if
X is not isomorphic to a Hilbert space then there always exists a bounded
sequence (Mk )k∈Z ⊂ L(X), which may even be chosen lacunary, such that
(Mk )k∈Z is not an L2 -multiplier.
This phenomenon had been discovered by G. Pisier (unpublished). We
want to explain this in more detail and give some extensions showing in
particular that Theorem 1.3 holds merely on Hilbert spaces if R-boundedness
is replaced by boundedness.
A sequence (Mk )k∈Z ⊂ L(X) is called lacunary if Mk = 0 for all
k ∈ Z \ {±2m : m ∈ N0 } ∪ {0}. We recall the following inequality due to
Pisier [Pi1]: for 1 ≤ p < ∞, there exist α, β > 0 such that
α
rj ⊗ xj ≤
e2j ⊗ xj j
p
p
j
L (0,1;X)
L (0,2π;X)
≤β
rj ⊗ xj p
j
L (0,1;X)
which holds for all xj ∈ X, where X is an arbitrary Banach space.
The operator-valued Marcinkiewicz multipler theorem
321
Recall that a Banach space X is of type 1 ≤ p ≤ 2 if, there exists C > 0
such that for x1 , x2 , · · · , xn ∈ X, we have
1/p

n
n
rj ⊗ xj ≤C
xj p  .
j=1
2
j=1
L (0,1;X)
X is of cotype 2 ≤ q ≤ ∞ if, there exists C > 0 such that for x1 , x2 , · · · ,
xn ∈ X, we have
n
n
xj q )1/q ≤ C rj ⊗ xj L2 (0,1;X) .
(
j=1
j=1
(with the usual modification if q = ∞) [Pi2] (see also [LT]). It is well known
that every Banach space is of type 1 and of cotype ∞, and for every measure
space (Ω, Σ, µ) and for every 1 ≤ p < ∞, the space Lp (Ω, Σ, µ) is of type
M in(2, p) and of cotype M ax(2, p). Kwapien has shown that a Banach
space X is isomorphic to a Hilbert space if and only if X is of type 2 and
of cotype 2 [Kw] (see also [LT, p. 73, 74]). Finally, a Banach space is said
to have a non trivial type if it is of type p for some 1 < p ≤ 2.
Proposition 1.12. Let X be a Banach space and 1 < p < ∞. Then the
following assertions are equivalent:
(i) X has a non trivial type;
(ii) for every Banach space Y , each lacunary R-bounded sequence in
L(X, Y ) defines an Lp -multiplier;
(iii) each lacunary R-bounded sequence in L(X) defines an Lp -multiplier.
Proof. (i) ⇒ (ii). Assume that X has a non trivial type and let Y be a
Banach space. By Lemma 6 of [Le] (see also [Pi2]), there exists C > 0 such
that for f ∈ Lp (0, 2π; X),
n
ˆ
n ⊗ f (2 )
e
≤ Cf Lp (0,2π;X)
2
n≥0
p
L (0,2π;X)
e−2n ⊗ fˆ(−2n )
≤ Cf Lp (0,2π;X) .
n≥0
p
L (0,2π;X)
Let (Mk )k∈Z ⊂ L(X, Y ) be a lacunary R-bounded sequence. By Pisier’s
inequality,
ek ⊗ Mk fˆ(k)
k∈Z
Lp (0,2π;Y )
322
W. Arendt, S. Bu
ˆ
ˆ
≤
e
⊗
M
+
e
⊗
M
f
(k)
f
(k)
k
k
k
k
k≥0
p
k<0
Lp (0,2π;Y )
L (0,2π;Y )


k
≤ β 
rk+1 ⊗ M2k fˆ(2 )
k≥0
p
L (0,1;Y )


k ˆ
+
r
⊗
M
k f (−2 )
 + M0 f Lp (0,2π;X)
k+1
−2
k≥0
p
L (0,1;Y )


k
≤ βRp (Mk : k ∈ Z) 
rk+1 ⊗ fˆ(2 )
k≥0
p
L (0,1;X)


k ˆ
+
rk+1 ⊗ f (−2 )
 + M0 f Lp (0,2π;X)
k≥0
p
L (0,1;X)


−1
k ˆ
≤ α βRp (Mk : k ∈ Z) 
e2k ⊗ f (2 )
k≥0
p
L (0,2π;X)


k ˆ
+
e−2k ⊗ f (−2 )
 + M0 f Lp (0,2π;X)
k≥0
p
L (0,2π;X)
≤ (2α
−1
βCRp (Mk : k ∈ Z) + M0 )f Lp (0,2π;X) .
This shows that (Mk )k∈Z is an Lp -multiplier.
(ii) ⇒ (iii) is trivial by taking Y = X.
(iii) ⇒ (i). Assume that every lacunary R-bounded sequence (Mk )k∈Z ⊂
L(X) defines an Lp -multiplier. Define Mk = I if k = 2n for some n ∈ N0
and Mk = 0 otherwise. Then (Mk )k∈Z is lacunary and R-bounded, by
assumption there exists C > 0 such that for f ∈ Lp (0, 2π; X),
n ˆ
n
e2 ⊗ f (2 )
n≥0
≤ C f Lp (0,2π;X) .
Lp (0,2π;X)
This implies that the closed subspace of Lp (0, 2π; X) generated by {e2n ⊗
xn : n ∈ N0 , xn ∈ X} is complemented in Lp (0, 2π; X). By Lemma 6 of
[Le] X has a non trivial type.
The operator-valued Marcinkiewicz multipler theorem
323
If X is isomorphic to a Hilbert space, then a subset T of L(X) is Rbounded if and only if it is bounded. Actually the following more general
proposition holds. The authors are indebted to C. Le Merdy and G. Pisier
for communicating them this result. We include the short proof for completeness.
Proposition 1.13. Let X and Y be Banach spaces. Then the following assertions are equivalent:
(i) X is of cotype 2 and Y is of type 2;
(ii) Each bounded subset in L(X, Y ) is R-bounded.
Proof. Assume that X is of cotype 2 and Y is of type 2. Let M ⊂ L(X, Y )
be a bounded subset and let C, C > 0 be the constants in the definitions
of type and cotype. Then for T1 , T2 , · · · , Tn ∈ M, x1 , x2 , · · · , xn ∈ X, we
have
n
n
rj ⊗ Tj xj ≤ C(
Tj xj 2 )1/2
2
j=1
j=1
L (0,1;Y )
1/2

n
≤ CsupT ∈M T 
xj 2 
j=1
n
≤ CC supT ∈M T rj ⊗ xj j=1
.
L2 (0,1;X)
This shows that M is R-bounded.
Conversely, assume that each bounded set in L(X, Y ) is R-bounded. Let
e ∈ X, e∗ ∈ X ∗ such that e, e∗ = e = e∗ = 1.
Then the set T = {e∗ ⊗ y : y ∈ Y , y ≤ 1} is R-bounded, by assumption.
y
Let y1 , . . . , ym ∈ Y . Then Tj = e∗ ⊗ yjj ∈ T. Hence
m
rj ⊗ y j j=1
L2 (0,1;Y )
m
=
rj ⊗ yj · Tj eL2 (0,1;Y )
j=1
m
≤ R2 (T) rj ⊗
yj · eL2 (0,1;X)
j=1
1

2
m
2
= R2 (T) 
yj  .
j=1
324
W. Arendt, S. Bu
This shows that Y is of type 2. In order to prove that X is of cotype 2, let
α := R2 ({x∗ ⊗ f : x∗ ∈ X ∗ , x∗ ≤ 1}) which is finite by assumption,
where f ∈ Y, f = 1 is fixed. Let x1 , . . . , xm ∈ X. Choose x∗j ∈ X ∗
such that x∗j = 1 and x∗j , xj = xj . Let Sj = x∗j ⊗ f . Then


m
j=1
1/2
xj 2 
m
=
rj ⊗ xj · f L2 (0,1;Y )
j=1
m
=
rj ⊗ Sj xj 2
j=1
L (0,1;Y )
m
≤ α
rj ⊗ xj .
2
j=1
L (0,1;X)
This proves that X is of cotype 2.
In view of Proposition 1.13 we may now formulate the following interesting special case of the Marcinkiewicz multiplier theorem.
Corollary 1.14. Let X = Lp1 (Ω, Σ, µ), Y = Lp2 (Ω, Σ, µ) where 1 <
p1 ≤ 2 ≤ p2 < ∞ and (Ω, Σ, µ) is a measure space. Then each bounded
sequence (Mk )k∈Z in L(X, Y ) satisfying supk∈Z k(Mk+1 − Mk ) < ∞
is an Lp -multiplier for each 1 < p < ∞.
Proposition 1.13 shows that in Proposition 1.12 R-boundedness may not
be replaced by boundedness, unless X is a Hilbert space. More precisely,
the following holds.
Proposition 1.15. Let X be a Banach space and 1 < p < ∞. The following
assertions are equivalent:
(i) X is isomorphic to a Hilbert space;
(ii) Each bounded lacunary sequence in L(X) is an Lp -multiplier.
Proof. (i) ⇒ (ii). Let X be a Banach space isomorphic to a Hilbert space
and (Mk )k∈Z be a bounded lacunary sequence. Then (Mk )k∈Z is R-bounded
by Proposition 1.13 as X is of type 2 and of cotype 2 (or by direct verification). Proposition 1.12 shows that the sequence is an Lp -multiplier.
(ii) ⇒ (i). It follows from the assumption and Proposition 1.11 that each
bounded sequence in L(X) is R-bounded. By Proposition 1.13 and
Kwapien’s Theorem, this implies that X is isomorphic to a Hilbert space.
The operator-valued Marcinkiewicz multipler theorem
325
Remark 1.16. Using Proposition 1.12 and the same argument as in the proof
of Proposition 1.15, we can easily establish the following: If X has a non
trivial type, then X is of cotype 2 and Y is of type 2 if and only if each
bounded lacunary sequence in L(X, Y ) is an Lp -multiplier.
The following proposition shows that we can not replace the R-boundedness in Theorem 1.3 by boundedness in operator norm unless the underlying
Banach spaces X is of cotype 2 and Y is of type 2 (when X = Y , this is
equivalent to say that X is isomorphic to a Hilbert space).
Proposition 1.17. Let X and Y be U M D-spaces. Then the following assertions are equivalent:
(i) X is of cotype 2 and Y is of type 2;
(ii) There exists 1 < p < ∞ such that each sequence (Mk )k∈Z ⊂ L(X, Y )
satisfying supk∈Z Mk < ∞ and supk∈Z k(Mk+1 − Mk ) < ∞ is
an Lp -multiplier.
Proof. (i) ⇒ (ii). Assume that X is of cotype 2 and Y is of type 2, then by
Proposition 1.13 each bounded subset in L(X, Y ) is actually R-bounded,
so the result follows from Theorem 1.3.
(ii) ⇒ (i). Assume that for some 1 < p < ∞, each sequence (Mk )k∈Z ⊂
L(X, Y ) satisfying supk∈Z Mk < ∞ and supk∈Z k(Mk+1 −Mk ) < ∞
defines an Lp -multiplier. Let (Mk )k≥0 ⊂ L(X, Y ) be a bounded sequence.
Define (Nn )n∈Z ∈ L(X, Y ) by


0
if n ≤ 0
Mk
if n = 2k for some k ≥ 0
Nn =

n−2k
Mk + 2k (Mk+1 − Mk ) if 2k ≤ n < 2k+1 for some k ≥ 0 .
Then one can easily verify that
sup Nn = sup Mk < ∞
n∈Z
k≥0
sup n(Nn+1 − Nn ) ≤ 4 sup Mk < ∞.
n∈Z
k≥0
Therefore the sequence (Nn )n∈Z is an Lp -multiplier by assumption. By
Proposition 1.11 this implies that the sequence (Nn )n∈Z is R-bounded, in
particular the sequence (Mk )k≥0 is R-bounded. We deduce from this that
each bounded subset in L(X, Y ) is actually R-bounded, By Proposition
1.13, this implies that X is of cotype 2 and Y is of type 2.
Finally, we remark that in the scalar case more general conditions are
known to be sufficient in Theorem 1.3. Let (Mk )k∈Z be a bounded scalar
326
W. Arendt, S. Bu
sequence. Instead of assuming that k(Mk+1 − Mk ) is bounded, it suffices
to assume that
|Mk+1 − Mk | < ∞
sup
j∈N
2j ≤|k|<2j+1
in order to deduce that (Mk )k∈Z is an Lp -multiplier for 1 < p < ∞ (see
[EG, Chapter 8]). This is the classical Marcinkiewicz multiplier theorem.
S̆trkalj and Weis [SW] give an operator-valued version of this result, where
the absolute value is replaced by a certain norm T which is defined as
the gauge function of an R-bounded set T in L(X). Actually, our Theorem
1.3 can be deduced from the results in [SW], but the proofs given there are
more complicated. They depend in particular on the work by Zimmermann
[Zi].
2. Strong Lp -well posedness of the periodic problem
We first introduce periodic Sobolev spaces. Let X be a Banach space.
Lemma 2.1. Let 1 ≤ p < ∞ and let u, u ∈ Lp (0, 2π; X). The following
are equivalent:
(i)
2π
0
u (t)dt = 0 and there exists x ∈ X such that
t
u(t) = x +
u (s)ds
a.e. on [0, 2π] ;
0
(ii) (u )ˆ(k) = ikû(k)
(k ∈ Z).
Proof. (i) ⇒ (ii). Let k ∈ Z \ {0}. Integration by parts yields,
1
û(k) =
2π
Since u (0) =
1
2π
2π
0
2π
−ikt
t
e
0
0
u (s)dsdt =
1 u (k) .
ik
u (s)ds = 0, assertion (ii) is proved.
(ii) ⇒ (i). Let v(t) =
above one has v̂(k) =
constant function.
t
u (s)ds. Since u (0) = 0 one has v(2π) = 0. As
0
1 ik u (k)
= û(k) for k ∈ Z \ {0}. Thus u − v is a
The operator-valued Marcinkiewicz multipler theorem
327
If u ∈ Lp (0, 2π; X), then there exists at most one u ∈ Lp (0, 2π; X)
such that the equivalent conditions of Lemma 2.1 are satisfied. We denote
1,p
by Hper
the space of all u ∈ Lp (0, 2π; X) such that there exists u ∈
Lp (0, 2π; X) such that the equivalent conditions of Lemma 2.1 are satisfied.
1,p
, it follows from (i) that u has a unique continuous representative.
If u ∈ Hper
We always identify u with this continuous function. Thus we have
t
u(t) = u(0) +
u (s)ds
(t ∈ [0, 2π])
0
1,p
and u(0) = u(2π) for all u ∈ Hper
.
1,p
Next we describe multipliers mapping Lp (0, 2π; X) into Hper
.
Lemma 2.2. Let 1 ≤ p < ∞. Let Mk ∈ L(X) (k ∈ Z). The following
assertions are equivalent:
(i) (Mk )k∈Z is an Lp -multiplier such that the associated operator M ∈
1,p
;
L(Lp (0, 2π; X)) maps Lp (0, 2π; X) into Hper
p
(ii) (kMk )k∈Z is an L -multiplier.
Proof. (i) ⇒ (ii). Let f ∈ Lp (0, 2π; X). By assumption, there exists g ∈
1,p
such that ĝ(k) = Mk fˆ(k) (k ∈ Z). Hence g (k) = ikMk fˆ(k)(k ∈
Hper
Z).
(ii) ⇒ (i). Let f ∈ Lp (0, 2π; X). By assumption, there exists v ∈
Lp (0, 2π; X) such that ikMk fˆ(k) = v̂(k) for all k ∈ Z. In particular,
2π
t
v(t)dt = 2πv̂(0) = 0. Let w(t) = v(s)ds. Then for k ∈ Z \ {0},
0
ŵ(k) =
1
ik v̂(k)
t
u(t) = x +
0
0
= Mk fˆ(k). Let u = w + e0 ⊗ (M0 fˆ(0) − ŵ(0)). Then
1,p
v(s)ds where x = u(0). Hence u ∈ Hper
. Moreover,
û(k) = ŵ(k) = Mk fˆ(k) for k ∈ Z \ {0} and û(0) = M0 fˆ(0).
Now let A be a closed operator on X. For 1 ≤ p < ∞, f ∈ Lp (0, 2π; X),
we consider the problem
Pper
u (t) = Au(t) + f (t) t ∈ (0, 2π)
u(0) = u(2π) .
1,p
By a strong Lp -solution we understand a function u ∈ Hper
such that
u(t) ∈ D(A) and u (t) = Au(t) + f (t) for almost all t ∈ [0, 2π].
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W. Arendt, S. Bu
Theorem 2.3 (Strong Lp -well-posedness). Assume that X is a U M Dspace. Let 1 < p < ∞. Then the following assertions are equivalent:
For each f ∈ Lp (0, 2π; X) there is a unique strong Lp -solution of
Pper ;
(ii) iZ ⊂ 8(A) and (kR(ik, A))k∈Z is an Lp -multiplier;
(iii) iZ ⊂ 8(A) and the sequence (kR(ik, A))k∈Z is R-bounded.
(i)
We say that Pper is strongly Lp -well-posed if these equivalent conditons
hold.
1,p
Proof. (i) ⇒ (ii). Let k ∈ Z. Let y ∈ X, f = ek ⊗y. There exists u ∈ Hper
such that u = Au + f . Taking Fourier transforms on both sides we obtain
that û(k) ∈ D(A) and ikû(k) = u (k) = Aû(k) + fˆ(k) = Aû(k) + y.
Thus (ik − A) is surjective. If (ik − A)x = 0, then u(t) = ek ⊗ x defines
a periodic solution of u = Au. Hence u = 0 by the assumption of uniqueness. We have shown that (ik − A) is bijective. Since A is closed we deduce
that ik ∈ 8(A).
Next we show that (kR(ik,A))k∈Z is an Lp -multiplier. Let f ∈ Lp (0, 2π; X).
1,p
such that u = Au + f .
By assumption, there exists a unique u ∈ Hper
Taking Fourier transforms, we deduce that û(k) ∈ D(A) and ikû(k) =
Aû(k) + fˆ(k); i.e., û(k) = R(ik, A)fˆ(k) for all k ∈ Z. Consequently,
u (k) = ikû(k) = ikR(ik, A)fˆ(k) for all k ∈ Z. This proves the claim.
1,p
(ii) ⇒ (i). Let f ∈ Lp (0, 2π; X). By Lemma 2.2 there exists u ∈ Hper
such
that û(k) = R(ik, A)fˆ(k) for all k ∈ Z. Since AR(ik, A) = ikR(ik, A) −
I, the sequence (AR(ik, A))k∈Z ⊂ L(X) is an Lp -multiplier. Observe that
A−1 is an isomorphism of X onto D(A) (seen as a Banach space with
the graph norm). Hence, (R(ik, A))k∈Z is an Lp -multiplier in L(X, D(A)).
This shows that u ∈ Lp (0, 2π; D(A)). Since u (k) = ikû(k) =
ikR(ik, A)fˆ(k) = AR(ik, A)fˆ(k) + fˆ(k) = Aû(k) + fˆ(k) for all k ∈ Z,
one has u = Au + f by the uniqueness theorem; i.e., u is a strong solution
1,p
of Pper . It remains to show uniqueness. If u ∈ Hper
∩ Lp (0, 2π; D(A))
such that u (t) = Au(t) (t ∈ (0, 2π)), then û(k) ∈ D(A) and ikû(k) =
Aû(k). Since ik ∈ 8(A) this implies that û(k) = 0 for all k ∈ Z and thus
u = 0.
(ii) ⇒ (iii) follows from Proposition 1.11.
(iii) ⇒ (ii). Let Mk = ikR(ik, A). We show that the set {k(Mk+1 −Mk ) :
k ∈ Z} is R-bounded. Then (ii) follows Theorem 1.3. One has
k(Mk+1 − Mk ) =
k(i(k + 1)R(i(k + 1), A) − ikR(ik, A)) =
ikR(i(k + 1), A)((k + 1)(ik − A) − k(i(k + 1) − A))R(ik, A) =
ikR((i(k + 1), A)(−A)R(ik, A) =
The operator-valued Marcinkiewicz multipler theorem
329
ikR(i(k + 1), A)(I − ikR(ik, A)) .
Since the product of R-bounded sequences is R-bounded, the claim follows.
Corollary 2.4. Let X be a U M D-space. If there exists p ∈ (1, ∞) such
that Pper is strongly Lp -well-posed then Pper is strongly Lp -well-posed for
all p ∈ (1, ∞).
We conclude this section mentioning analogue non-discrete results. In
one of the first investigation on maximal regularity, Mielke [Mi] studies
strong Lp -well-posedness on the entire line. This leads to ”bisectorial operators” which are not necessarily generators of C0 -semigroups. Mielke
proves p-independance of maximal regularity and gives a characterization
on Hilbert spaces. Further results on U M D-spaces are obtained recently by
Schweiker [Sch].
3. Mild solutions
Let A be a closed operator on a Banach space X. Let f ∈ L1 (0, 2π; X).
A function u ∈ C([0, 2π]; X) is called a mild solution of the problem Pper
(see Sect. 2) if u(0) = u(2π) and
 t


 u(s)ds ∈ D(A) and
0
(3.1)
t
t


 u(t) − u(0) = A u(s)ds + f (s)ds
0
0
Lp -solution
is a mild solufor all t ∈ [0, 2π]. It is clear that every strong
1,p
, then u is a strong
tion. Conversely, if u is a mild solution and u ∈ Hper
Lp -solution. We want to describe mild solutions in terms of the Fourier coefficients.
For this we need the following lemma.
Lemma 3.1. Let f, g ∈ Lp (0, 2π; X), where 1 ≤ p < ∞. Then the following are equivalent.
(i) f (t) ∈ D(A) and Af (t) = g(t) a.e.;
(ii) fˆ(k) ∈ D(A) and Afˆ(k) = ĝ(k) for all k ∈ Z.
Proof. (i) ⇒ (ii). This follows from the closedness of A (cf. [ABHN,
Proposition 1.1.7]).
(ii) ⇒ (i). There exists n converging to ∞ as & → ∞ such that σn (f )(t) →
f (t) a.e. and σn (g)(t) → g(t) a.e. as & → ∞ (cf. (1.1)). Since σn (f )(t) ∈
D(A) and Aσn (f )(t) = σn (g)(t) for all t ∈ [0, 2π], the claim follows
from the closedness of A.
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W. Arendt, S. Bu
Proposition 3.2. Let u ∈ C([0, 2π], X) such that u(0) = u(2π). Assume
that D(A) = X. Then u is a mild solution of Pper if and only if
(3.2)
û(k) ∈ D(A) and (ik − A)û(k) = fˆ(k) for all k ∈ Z .
Proof. 1. Assume that u is a mild solution. Letting t = 2π in (3.1) we see that
t
û(0) ∈ D(A) and −Aû(0) = fˆ(0). Consider the functions v(t) = u(s)ds
0
t
and g(t) = u(t) − u(0) − f (s)ds. Then by Lemma 3.1, v̂(k) ∈ D(A) and
0
1
1
Av̂(k) = ĝ(k) for all k ∈ Z. For k = 0 we have v̂(k) = − ik
û(0) + ik
û(k),
1 ˆ
1 ˆ
ˆ
and ĝ(k) = û(k) + ik f (0) − ik f (k). Since −Aû(0) = f (0) we obtain
(3.2).
2. Conversely, assume that (3.2) holds. Let x∗ ∈ D(A∗ ). By [ABHN, Proposition B.10], it suffices to show that
t
∗ ∗
∗
∗
t
u(s) , A x ds = u(t) , x − u(0) , x −
0
f (s) , x∗ ds .
0
+ f (s), x∗ . Then ŵ(k) =
t
ikû(k), x∗ for all k ∈ Z by assumption (3.2). Consider g(t) = w(s)ds−
Consider the function w(s) = u(s),
A∗ x∗ 0
1
1
u(t), x∗ . Then for k ∈ Z \ {0}, ĝ(k) = − ik
ŵ(0) + ik
ŵ(k) − û(k),
x∗ = 0 since ŵ(0) = 0. It follows from the Uniqueness Theorem that g is
constant; i.e. g(t) = g(0) = −u(0), x∗ for all t ∈ [0, 2π]. This is precisely
what we claimed.
As a corollary we obtain the following characterization of uniqueness of
mild solutions of Pper . By σp (A) we denote the set of all eigenvalues of A.
Corollary 3.3. The following assertions are equivalent:
(i) For all f ∈ L1 (0, 2π; X) there exists at most one mild solution of Pper ;
(ii) iZ ∩ σp (A) = ∅.
Next we want to characterize well-posedness of Pper in the mild sense.
Proposition 3.4. Assume that D(A) = X. Let 1 ≤ p < ∞. Assume that
for all f ∈ Lp (0, 2π; X) there exists a unique mild solution of Pper . Then
iZ ⊂ 8(A) and (R(ik, A))k∈Z is an Lp -multiplier.
Proof. As in the proof of Theorem 2.3 one sees that iZ ⊂ 8(A). Let f ∈
Lp (0, 2π; X). Let u be the mild solution of Pper . It follows from (3.2) that
û(k) = R(ik, A)fˆ(k) for all k ∈ Z. Now the claim follows from Proposition
1.1.
The operator-valued Marcinkiewicz multipler theorem
331
We do not know whether the converse of Proposition 3.4 is true in general.
Given f , u ∈ Lp (0, 2π; X) such that û(k) = R(ik, A)fˆ(k) (k ∈ Z), the
problem is to show that u is continuous.
This is the case if A generates a C0 -semigroup. In that case, mild solutions
can be described differently [ABHN, Proposition 3.1.16]:
Lemma 3.5. Let T be a C0 -semigroup with generator A. Let τ > 0,
f ∈ L1 (0, τ ; X), u ∈ C([0, τ ],X), x ∈ X. The following assertions are
equivalent:
t
(i)
u(s)ds ∈ D(A) and
0
t
t
u(s)ds +
u(t) − x = A
0
(ii) u(t) = T (t)x + T ∗ f (t)
f (s)ds a.e.
0
(t ∈ [0, τ ]), where T ∗ f (t) =
s)f (s)ds.
t
0
T (t −
Now we obtain the following characterization of mild Lp -well-posedness.
Theorem 3.6. Let A be the generator of a C0 -semigroup T and let 1 ≤ p <
∞. Then the following are equivalent:
(i) For all f ∈ Lp (0, 2π; X) there exists a unique mild solution u of Pper ;
(ii) iZ ⊂ 8(A) and (R(ik, A))k∈Z is an Lp -multiplier;
(iii) 1 ∈ 8(T (2π)).
Proof. (i) ⇒ (ii) is Proposition 3.4.
(ii) ⇒ (i) . It will be convenient to identify Lp (0, 2π; X) with Lp2π (R, X)
of all 2π-periodic X-valued functions f such that the restriction of f on
m
n
1
[0, 2π] is p-integrable. Let f ∈ Lp2π (R, X), fn = n+1
ek ⊗ fˆ(k).
m=0 k=−m
Then fn ∈ C ∞ (R, X) is 2π-periodic and lim fn = f in Lp (0, 2π; X).
n→∞
m
n
1
ˆ
ek ⊗ R(ik, A)f (k). The hypothesis implies that
Let un = n+1
m=0 k=−m
u = lim un exists in Lp (0, 2π; X). The function un is in C ∞ (R, D(A))
n→∞
and un (t) = Aun (t) + fn (t) (t ∈ R). We find a subsequence such that
un (r) → u(r) and fn (r) → f (r) a.e. as & → ∞. Fix r0 ≤ 0 such that
lim un (r0 ) = u(r0 ). Let vn (t) = un (t + r0 ). Then vn (t) = Avn (t) +
→∞
fn (t + r0 ). It follows that
t
vn (t) = T (t)un (r0 ) +
T (s)fn (t + r0 − s)ds
0
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W. Arendt, S. Bu
for all t ≥ 0. The right hand side converges to v(t) := T (t)u(r0 ) +
t
T (s)f (t + r0 − s)ds in X for all t ≥ 0 as n → ∞. Observe that
0
v : R+ → X is continuous. Since vn is 2π-periodic, also v is 2π-periodic.
Since un → u a.e. we have u(t+r0 ) = v(t) (t ≥ 0) a.e. Changing u on a
set of measure 0, we may assume that u(t + r0 ) = v(t) for all t ≥ 0. In particular, taking t = −r0 , we have u(0) = lim vn (−r0 ) = lim un (0).
→∞
→∞
Thus we may choose r0 = 0 in the above argument and deduce that
t
u(t) = T (t)u(0) + T (s)f (t − s)ds. Since u(2π) = u(0), it follows
0
from Lemma 3.5 that u is a mild solution of Pper . Uniqueness of the solution follows from (3.2) by the Uniqueness Theorem.
(iii) ⇒ (i). Let f ∈ Lp (0, 2π; X). Choose x = (I − T (2π))−1 (T ∗ f )(2π)
and u(t) = T (t)x + (T ∗ f )(t). Then u(0) = u(2π) and u is a mild solution
of Pper by Lemma 3.5.
(i) ⇒ (iii) follows from Prüss [Pr].
Next we show that the condition in Theorem 3.6 cannot be replaced by
the weaker condition that (R(is, A))s∈R be R-bounded. In other words, the
well-known characterization of negative type on Hilbert space by boundedness of the resolvent on the right half plane [ABHN, Theorem 5.2.1] due to
Prüss [Pr] is not true on Lp -spaces for p = 2 even if boundedness is replaced
by the stronger assumption of R-boundedness.
Example 3.7. There exists the generator A of a C0 -semigroup T on the
space X = Lp (0, ∞), where 2 < p < ∞, such that s(A) := sup{Reλ :
λ ∈ σ(A)} < 0 and such that the set {R(λ, A) : Reλ ≥ 0} is R-bounded .
But (R(ik, A))k∈Z is not an Lq -multiplier for any q ∈ [1, ∞).
Proof. Let 2 < p < ∞. In [Ar] (see also [ABHN, Example 5.1.11]) a
positive C0 -semigroup T on Y := Lp (0, ∞) ∩ L2 (0, ∞) is constructed
whose generator A satisfies s(A) < 0 but T has type ω(T ) = 0. Since
T is positive, this implies that 1 = etω(T ) = r(T (t)) ∈ σ(T (t)). Thus
{R(ik, A) : k ∈ Z} is not an Lq -multiplier for any q ∈ [1, ∞). We show
that still, {R(ik, A) : k ∈ Z} is R-bounded. In fact, by [LT, Remark on p.
191 and Section 2.f], the space Y is isomorphic to Lp (0, ∞) as a Banach
space. By [LT, 1.d.7 (ii)], it follows that Y is a p-concave Banach lattice.
∞
Since R(λ, A)f = e−λt T (t)f dt one has
0
|R(λ, A)f | ≤ R(0, A)|f | for all f ∈ Y
whenever Reλ ≥ 0. Now it follows from Maurey’s result [LT, Theorem
1.d.6] that the set {R(λ, A) : Reλ ≥ 0} is R-bounded. Since Y is isomorphic to Lp (0, ∞) all claims are proved.
The operator-valued Marcinkiewicz multipler theorem
333
4. Hölder continuous solution
In this section we show that Pper has a unique Hölder continuous solution
whenever the resolvent decreases fast enough on the imaginary axis. For 0 <
α < 1 we denote by C α ([0, 2π]; X) the space of all continuous functions
f : [0, 2π] → X such that
f (t) − f (s) ≤ c|t − s|α
(s, t ∈ [0, 2π])
for some c ≥ 0. The following is the main result of this section.
Theorem 4.1. Let A be a closed operator on a U M D-space X such that
iZ ⊂ 8(A). Assume that
sup |n|θ R(in, A) < ∞
(4.1)
n∈Z
1
where 3/4 < θ ≤ 1. Let 4θ−3
< p < ∞, 0 < α < 4θ − 3 − p1 . Then
p
for each f ∈ L (0, 2π; X) there exists a unique mild solution u of Pper .
Moreover, u ∈ C α ([0, 2π]; X).
We need the following lemma.
Lemma 4.2. Let 8 > 0, λ1 > 0, 0 ≤ θ ≤ 1. Define inductively λn+1 =
∞
1
converges.
λn + 8λθn . Then for γ > 1 − θ the series
λγ
n=1
n
Proof. Let α > 0 such that
γ − (1 − θ)
α
θ+γ−1
=
.
>
γ
γ
1+α
Let δ > 0, ε > 0 such that γ8 − ε > δ(1 + α) + ε. Let for n ∈ N,
An = λγn
θ+γ−1
γ
Bn+1 = Bn + (γ8 − ε)Bn
,
B1 = 1
Cn+1 = Cn + (δ(1 + α) + ε)Cn
Dn = (1 + nδ)1+α .
, C1 = 1
α
1+α
Then
lim An = lim Bn = lim Cn = lim Dn = ∞ .
n→∞
n→∞
n→∞
n→∞
334
W. Arendt, S. Bu
One has
Dn+1
1+α
δ
1+
= (1 + nδ)
1 + nδ
1
δ(1 + α)
+o
= (1 + nδ)1+α 1 +
1 + nδ
1 + nδ
α
α
= Dn + δ(1 + α)Dn1+α + o Dn1+α
1+α
α
≤ Dn + (δ(1 + α) + ε)Dn1+α
when n ≥ n0 for some n0 ∈ N. Let n1 ∈ N such that Dn0 ≤
C1n1 . One
shows by induction that Dn0 +m ≤ Cn1 +m for all m ∈ N. Since
Dn < ∞
∞
1
we conclude that
Cn < ∞. Choose n2 ∈ N such that C1 ≤ Bn2 . Since
n=1
(δ(1 + α) + ε) ≤ γ8 − ε and
for all m ∈ N. Consequently
α
1+α
∞
m=1
≤
θ+γ−1
γ
1
Bm
it follows that C1+m ≤ Bn2 +m
< ∞. Similarly as above one has
γ
An+1 = λγn (1 + 8λθ−1
n )
= λγn (1 + 8γλθ−1
+ o(λθ−1
n
n ))
θ+γ−1
γ
≥ An + (γ8 − ε)An
for n ≥ n3 if n3 is large enough. Choose n4 such that
1 ≤ An4 . Then it
B
1
follows that Bn+1 ≤ Am+n4 for all m ∈ N. Since
Bm < ∞, it follows
1
that
Am < ∞.
Proposition 4.3. Let s0 ≥ 1, 1/2 < θ ≤ 1. Assume that {is : s ∈ R , |s| ≥
s0 } ⊂ 8(A) and
sup |s|θ R(is, A) < ∞ .
(4.2)
|s|≥s0
Then the set {|s|β R(is, A) : |s| ≥ s0 } is R-bounded whenever 0 < β <
2θ − 1.
Proof. Let c ≥ 1 be larger than the supremum in (4.2). Let λ0 ≥ s0 such
1 θ
that λ0 − 2c
λ0 ≥ s0 . By Taylor’s formula we have
R(iλ, A) = R(iλ0 , A)
∞
(iλ0 − iλ)k R(iλ0 , A)k
k=0
whenever λ ∈ I(λ0 ) := [λ0 −
1 θ
2c λ0 ,
Rq {λθ R(iλ, A) : λ ∈ I(λ0 )}
λ0 +
1 θ
2c λ0 ].
Hence for 1 ≤ q < ∞,
The operator-valued Marcinkiewicz multipler theorem
≤ R(iλ0 , A)
≤
c
λθ0
∞
∞
335
Rq {λθ R(iλ0 , A)k (λ − λ0 )k : λ ∈ I(λ0 )}
k=0
R(iλ0 , A)k 2 sup{|λ|θ |λ − λ0 |k : λ ∈ I(λ0 )}
k=0
∞
1 θ θ 1 θ k
c c k
λ
( ) · 2 λ0 + λ0
≤ θ
2c
2c 0
λ0 k=0 λθ0
θ
1
1 θ
= 4c 1 + λ0θ−1 ≤ 4c 1 +
≤ 8c .
2c
2c
1 θ
λn . Then
Define λn inductively by λn+1 = λn + 2c
Rq {|s|θ R(is, A) : s ∈ [λn , λn+1 ]}
Rq {|s|β R(is, A) : s ≥ λ0 } ≤
n≥0
· 2
sup
s∈[λn λn+1 ]
|s|β−θ ≤ 16c
1
θ−β
n≥0 λn
<∞
by Lemma 4.2 since θ − β > 1 − θ. The estimate for s ≤ −λ0 is similar.
Proof of Theorem 4.1. a) Using Taylor’s formula (4.2) one sees that s ∈ 8(A)
and R(is, A) ≤ C1 whenever |s| ≥ k0 where k0 ∈ N, C1 ≥ 0 are
suitable. Let s ≥ k0 . Choose s ∈ [k, k + 1]. Then
sθ R(is, A) − k θ R(ik, A) =(sθ − k θ )R(is, A) + k θ (R(is, A) − R(ik, A))
=(sθ − k θ )R(is, A)
+k θ R(ik, A)R(is, A)i(k − s) .
Similar for s ≤ −k0 . This shows that
C := sup |s|θ R(is, A) < ∞ .
|s|≥k0
b) Let 0 ≤ β < 4θ − 3. It follows from Proposition 4.3 that the set
(4.3)
{|s|
β+1
2
R(is, A) : |s| ≥ k0 }
is R-bounded. It follows from Lemma 1.8 that also {|s|β+1 R(is, A)2 :
|s| ≥ k0 } is R-bounded. Since β < 2θ − 1, also {|s|β R(is, A) : |s| ≥ k0 }
is R-bounded by Proposition 4.3. Let M (s) = sβ R(is, A) (|s| > k0 ).
Then sM (s) = βM (s) − isβ+1 R(is, A)2 . Hence {M (s) : |s| ≥ k0 } and
k+1
{sM (s) : |s| ≥ k0 } are R-bounded. Since Mk+1 − Mk =
M (s)ds,
k
336
W. Arendt, S. Bu
the set {k(Mk+1 − Mk ) : k ∈ Z; |k| ≥ k0 } is contained in co{sM (s) :
|s| ≥ k0 } and so R-bounded by Lemma 1.9. Theorem 1.3 implies that
(Mk )k∈Z is an Lp -multiplier.
c) Let f ∈ Lp (0, 2π; X). Applying b) to β0 = 0 and 0 < β < 4θ − 3 we
find unique functions u, v ∈ Lp (0, 2π; X) such that
û(k) = R(ik, A)fˆ(k) , v̂(k) = (ik)β R(ik, A)fˆ(k)
β,p
for all k ∈ Z. Thus u ∈ Hper
:= {w ∈ Lp (0, 2π; X): there exists g ∈
p
L (0, 2π; X) satisfying ĝ(k) = (ik)β ŵ(k) for all k ∈ Z}. Now we choose
1/p < β < 4θ − 3. Then by [Zy, Theorem 9.1, p. 138] one has
β,p
⊂ {w ∈ C
Hper
β− p1
([0, 2π]; X) : w(0) = w(2π)} .
5. Maximal regularity
In this section we compare the periodic problem Pper with the first order
problem with Dirichlet boundary condition
u (t) = Au(t) + f (t) (t ∈ [0, τ ])
P0 (τ )
u(0) = 0 ,
where A is the generator of a C0 -semigroup T and f ∈ L1 (0, τ ; X), τ > 0.
There exists a unique mild solution u = T ∗ f (see Lemma 3.5).
We say that P0 (τ ) is strongly Lp -well-posed if for every f ∈ Lp (0, τ ; X)
one has T ∗ f ∈ H 1,p (0, τ ; X).
It is easy to see that strong Lp -well-posedness of P0 (τ ) implies the same
property if A is replaced by A − λ for all λ ∈ C. Moreover, it is well-known
that Lp -well-posedness of P0 (τ ) for some τ > 0 implies the same property
for P0 (τ ) for all τ > 0 (see Dore [Do]).
Theorem 5.1. Let A be the generator of a C0 -semigroup T on a Banach
space X. Let 1 < p < ∞. The following assertions are equivalent:
(i) P0 (2π) is strongly Lp -well-posed and 1 ∈ 8(T (2π));
(ii) Pper is strongly Lp -well-posed.
Proof. If P0 (2π) is Lp -well-posed, then T is holomorphic (see Dore [Do]).
Conversely, it is not difficult to see from the necessity of condition (iii) in
Theorem 2.3 (for which the U M D-property is not needed) that Lp -wellposedness of Pper implies that T is holomorphic. Thus, for the proof of
The operator-valued Marcinkiewicz multipler theorem
337
equivalence of (i) and (ii) we can assume that T is holomorphic. By the
trace theorem we have
(X, D(A))1− 1 ,p := {x ∈ X : AT (·)x ∈ Lp (0, 2π; X)}
p
= {u(0) : u ∈ Lp (0, 2π; D(A)) ∩ H 1,p (0, 2π; X)} ,
see Lunardi [Lu, 1.2.2 and 2.2.1].
(i) ⇒ (ii). Let f ∈ Lp (0, 2π; X). Then by assumption v = T ∗ f ∈
H 1,p (0, 2π; X). It follows from the trace theorem above that v(2π) ∈
(X, D(A))1−1/p,p . Hence also
x := (I − T (2π))−1 v(2π) ∈ (X, D(A))1−1/p,p .
d
T (t)x = AT (t)x on (0, ∞), it follows that T (·)x ∈ H 1,p (0, 2π; X).
Since dt
Let u(t) = T (t)x + v(t). Then u ∈ H 1,p (0, 2π; X) and u(0) = x =
T (2π)x + v(2π) = u(2π). Thus u is a strong solution of Pper . Since
e2πσ(A) ⊂ σ(T (2π)) and 1 ∈ 8(T (2π)), it follows that iZ ⊂ 8(A), and
uniqueness of the solution of Pper follows from Corollary 3.3.
1,p
(ii) ⇒ (i). Let f ∈ Lp (0, 2π; X). By assumption, there exists v ∈ Hper
solution of Pper . It follows from the trace theorem again that x := v(0) ∈
(X, D(A))1−1/p,p ; hence T (·)x ∈ H 1,p (0, 2π; X). Let u(t) = v(t) −
T (t)x. Then u is a strong Lp -solution of P0 (2π).
With the help of Theorem 1.3 we now obtain the following characterization of strong Lp -well-posedness of P0 (τ ).
Corollary 5.2. Let A be the generator of a C0 -semigroup on a U M D-space
X and let 1 < p < ∞. The following assertions are equivalent:
(i) P0 (τ ) is strongly Lp -well-posed for all τ > 0;
(ii) there exists w > ω(T ) such that {kR(w + ik, A) : k ∈ Z} is Rbounded.
Proof. Replacing A by A − ω this follows directly from Theorem 5.1 and
Theorem 2.3
Corollary 5.2 shows in particular that strong Lp -well-posedness of P0 (τ )
is independent of p ∈ (1, ∞) (which is well-known). It became customary to
say that a closed operator A has the property (M R) (for maximal regularity)
if P0 (τ ) is strongly Lp -well-posed for one and hence all p ∈ (1, ∞), τ > 0.
Thus condition (ii) is a characterization of (M R).
We obtain this characterization as a consequence of the discrete multiplier theorem (Theorem 1.3). It is also possible to use Weis’ multiplier
theorem [W1, Theorem 3.4]) and the criterion [W2, Section 1e)(i)]. For
338
W. Arendt, S. Bu
that one has to show that condition (ii) of Corollary 5.2 implies that the set
{sR(is + ω, A) : s ∈ R} is R-bounded. This is not difficult to do.
Finally, we should mention that in contrast to the periodic problem, in
the context of problem P0 (τ ) it is natural to assume that A generates a
holomorphic C0 -semigroup. In fact, for densely defined closed operators
this is a necessary assumption (see Dore [Do]). By a spectacular result of
Kalton and Lancien [KL] it is not sufficient if X is a Banach space with
unconditional basis, which is not isomorphic to a Hilbert space.
6. The second order problem
Let A be a closed operator on a U M D-space X and let 1 < p < ∞. In this
section we characterize strong Lp -well-posedness of the problem
u (t) + Au(t) = f (t)
on bounded intervall with periodic, Dirichlet and Neumann boundary conditions. For a < b we denote by
H 2,p (a, b; X) := {u ∈ H 1,p (a, b; X) : u ∈ H 1,p (a, b; X)}
the second Sobolev space. Note that H 2,p (a, b; X) ⊂ C 1 ([a, b]; X). Using
the notion of Sect. 2 we let
2,p
1,p
1,p
Hper
:= {u ∈ Hper
: u ∈ Hper
}.
2,p
Let u ∈ Lp (0, 2π; X). It is easy to see that u ∈ Hper
if and only if there
exists v ∈ Lp (0, 2π; X) such that v̂(k) = −k 2 û(k) for all k ∈ Z. In that
case v = (u ) =: u .
Theorem 6.1. The following are equivalent:
(i) For all f ∈ Lp (0, 2π; X) there exists a unique
u ∈ Lp (0, 2π; D(A)) ∩ H 2,p (0, 2π; X)
such that
P2 (2π)
u (t) + Au(t) = f (t) a.e.
u(0) = u(2π) , u (0) = u (2π) ;
(ii) one has k 2 ∈ 8(A) for all k ∈ Z and {k 2 R(k 2 , A) : k ∈ Z} is
R-bounded.
The operator-valued Marcinkiewicz multipler theorem
339
Proof. (i) ⇒ (ii). One shows as in Theorem 2.3 that k 2 ∈ 8(A) for all k ∈
Z. Let f ∈ Lp (0, 2π; X). Let u be the solution of (i). Then û(k) ∈ D(A) and
−k 2 û(k) + Aû(k) = fˆ(k). Hence û(k) = −R(k 2 , A)fˆ(k) and (u )ˆ(k) =
−k 2 û(k) = k 2 R(k 2 , A)fˆ(k) (k ∈ Z). Since u ∈ Lp (0, 2π; X) this
proves (ii).
(ii) ⇒ (i). Let Mk = k 2 R(k 2 , A) (k ∈ Z). Then
k(Mk+1 − Mk ) = kR((k + 1)2 , A){(k + 1)2 (k 2 − A)
−k 2 ((k + 1)2 − A)}R(k 2 , A)
= −k(2k + 1)R((k + 1)2 , A)(k 2 R(k 2 , A) − I) .
It follows that the set {k(Mk+1 − Mk ) : k ∈ Z} is R-bounded. Now
Theorem 1.3 implies that (k 2 R(k 2 , A))k∈Z is an Lp -multiplier. Let f ∈
Lp (0, 2π; X). Then there exists u ∈ Lp (0, 2π; X) such that (u )ˆ(k) =
k 2 R(k 2 , A)fˆ(k) (k ∈ Z). A simple computation shows that there exist
t
y, z ∈ X such that if we let u(t) = (t − s)u (s)ds + ty + z for t ∈ [0, 2π],
0
then û(k) = −R(k 2 , A)fˆ(k) for all k ∈ Z.
Since AR(k 2 , A) = k 2 R(k 2 , A) − I, it follows that (R(k 2 , A))k∈Z is an
L(X, D(A))-multiplier. Thus u ∈ Lp (0, 2π; D(A)). Since
(u + Au)ˆ(k) = k 2 R(k 2 , A)fˆ(k) − AR(k 2 , A)fˆ(k)
= fˆ(k) (k ∈ Z)
it follows from the Uniqueness Theorem that u is a solution of P2 (2π). Since
(u )ˆ(0) = (u )ˆ(0) = 0 it follows that u (0) = u (2π) and u(0) = u(2π).
Uniqueness is proved as in Sect. 2.
In order to treat Dirichlet boundary conditions we will consider odd
functions f on (−π, π); i.e. functions satisfying fˆ(k) = −fˆ(−k) for all
k ∈ Z. We need the following lemma.
Lemma 6.2. Let Mk ∈ L(X) such that Mk = M−k (k ∈ Z). Assume
that for each odd f ∈ Lp (−π, π; X) there exists u ∈ Lp (−π, π; X) such
that û(k) = Mk fˆ(k) (k ∈ Z). Then (Mk )k∈Z is an Lp -multiplier.
Proof. Let f ∈ Lp (−π, π; X). We have to show that there exists g ∈
Lp (−π, π; X) such that Mk fˆ(k) = ĝ(k) for all k ∈ Z. We can assume
that fˆ(0) = 0. Consider f1 ∈ Lp (−π, π; X) such that

 fˆ(k) if k > 0
ˆ
f1 (k) = −fˆ(−k) if k < 0

0
if k = 0 .
340
W. Arendt, S. Bu
Notice that f1 ∈ Lp (−π, π; X) exists as X is a U M D-space [Bur]. There
exists h1 ∈ Lp (−π, π; X) such that Mk fˆ1 (k) = ĥ1 (k) (k ∈ Z). Since the
Riesz projection is bounded we find g1 ∈ Lp (−π, π; X) such that ĝ1 (k) =
ĥ1 (k) for k ≥ 0 and ĝ1 (k) = 0 for k < 0. Thus ĝ1 (k) = Mk fˆ(k) for k ≥ 0.
Similarly, we find g2 ∈ Lp (−π, π; X) such that ĝ2 (k) = Mk fˆ(k) for k < 0
and ĝ2 (k) = 0 for k ≥ 0. Choose g = g1 + g2 .
Now we obtain the following characterization of strong Lp -well-posedness in the case of Dirichlet boundary conditions. Here 0 may be in the
spectrum of A.
Theorem 6.3. The following are equivalent:
(i) For all f ∈ Lp (0, π; X) there exists a unique u ∈ Lp (0, π; D(A)) ∩
H 2,p (0, π; X) satisfying
u (t) + Au(t) = f (t) a.e.
u(0) = u(π) = 0
(ii) k 2 ∈ 8(A) for all k ∈ N and {k 2 R(k 2 , A) : k ∈ N} is R-bounded.
Proof. (i) ⇒ (ii). Let k ∈ N. We show that k 2 ∈ 8(A). If x ∈ D(A)
such that (−k 2 + A)x = 0, then u(t) = (sin kt)x defines a solution of
u + Au = 0. Hence u = 0, and so x = 0. Let y ∈ X. There exists a strong
solution u of u + Au = (sin kt)y. Extend u to an odd function. Comparing
Fourier coefficients we see that u(t) = (sin kt) · x for some x ∈ D(A)
satisfying (−k 2 + A)x = y. We have shown that (−k 2 + A) is bijective,
thus k 2 ∈ 8(A).
Let f ∈ Lp (0, π; X). There exists a unique function u satisfying (i).
Extending u and f to odd functions we see that −k 2 û(k) + Aû(k) =
fˆ(k), hence û(k) = −R(k 2 , A)fˆ(k) for all k ∈ Z. Moreover, (u )ˆ(k) =
−k 2 R(k 2 , A)fˆ(k) (k ∈ Z). Now (ii) follows from Lemma 6.2.
(ii) ⇒ (i). Let M0 = 0, Mk = k 2 R(k 2 , A) for k ∈ Z \ {0}. One
sees as in the proof of Theorem 6.1 that (Mk )k∈Z is an Lp -multiplier.
Let f ∈ Lp (0, π; X). Extend f to an odd function. Then there exists
u ∈ Lp (−π, π; X) such that (u )ˆ(k) = k 2 R(k 2 , A)f(k) for k = 0 and
(u )ˆ(0) = 0. A simple computation shows that there exists x ∈ X such
t
that if we let u(t) = (t − s)u (s)ds + tx for t ∈ [0, π] and extend u to an
0
odd function on [−π, π], then û(k) = −R(k 2 , A)fˆ(k) for k = 0. So u|[0,π]
solves the problem in (i).
Finally we consider Neumann boundary conditions.
Theorem 6.4. The following assertions are equivalent:
The operator-valued Marcinkiewicz multipler theorem
341
(i) For all f ∈ Lp (0, π; X) there exists a unique u ∈ Lp (0, π; D(A)) ∩
H 2,p (0, π; X) satisfying
u (t) + Au(t) = f (t) a.e.
u (0) = u (π) = 0 ;
(ii) one has k 2 ∈ 8(A) for all k ∈ N0 and {k 2 R(k 2 , A) : k ∈ N} is
R-bounded.
The proof may be given similarly to the one of Theorem 6.3 replacing
odd by even functions there and in Lemma 6.2.
Finally we mention that Clément and Guerre-Delabrière [CG] studied
the relation of first and second order problems. To be more precise, let B be
a closed operator and consider the periodic problem
u + Bu = f
Pper
u(0) = u(2π)
of Sect. 2. Let A = −B 2 . Let 1 < p < ∞. Assume that Pper is strongly Lp well-posed. Then by Theorem 2.3 we have iZ ⊂ 8(−B) and {k(ik +B)−1 :
k ∈ Z} is R-bounded. Then k 2 ∈ 8(A) and R(k 2 , A) = (k 2 + B 2 )−1 =
(ik +B)−1 (−ik +B)−1 for all k ∈ Z. It follows that {k 2 R(k 2 , A) : k ∈ Z}
is R-bounded and Theorem 6.1, 6.3 and 6.4 give strong Lp -well-posedness of
the second order problems defined by A. This is shown in [CG] by different
methods in the case when −B generates an exponentially stable holomorphic
C0 -semigroup T . In that case they also show the other implication. From
our results this other implication can be seen as follows. One may represent
the resolvent of B by the resolvent of B 2 via a contour integral [Ta, (2.29)
page 36]. If the equivalent conditions appearing in Theorem 6.1, 6.3 or
6.4 are satisfied, then it is not difficult with help of this formula to prove
R-boundedness of {k(ik − B)−1 : k ∈ Z} which implies strong Lp -wellposedness of Pper by Theorem 2.3 again.
References
[ABHN]
[Ar]
[Bo]
W. Arendt, C. Batty , M. Hieber, F. Neubrander: Vector-valued Laplace
Transforms and Cauchy Problems. Monographs in Mathematics. Birkhäuser,
Basel, 2001, ISBN 3-7643-6549-8.
W. Arendt: Spectrum and growth of positive semigroups. Evolution Equations. Baton Rouge 1992. G. Ferreyra, G. Goldstein, F. Neubrander eds:
Marcel Dekker, New York (1994), 21–24.
J. Bourgain: Vector valued singular integrals and the H 1 − BM O duality. In
Burkholder, editor, Probability Theory and Harmonic Analysis, pages 1–19,
New York, 1986. Marcel Dekker.
342
[Bur]
[CG]
[CP]
[CPSW]
[DG]
[Do]
[DV]
[EG]
[KL]
[Kw]
[LS]
[Le]
[LT]
[Lu]
[Mi]
[Ne]
[Pi1]
[Pi2]
[Pr]
[Sch]
[SW]
[Ta]
W. Arendt, S. Bu
D. Burkholder: A geometric condition that implies the existence of certain
singular integrals of Banach-space-valued functions. Proc. of Conf. on Harmonic Analysis in Honor of Antoni Zygmund, Chicago 1981, Wadsworth
Publishers, Belmont, Calif. 1983, 270–286.
Ph. Clément, S. Guerre-Delabrière: On the regularity of abstract Cauchy
problems of order one and boundary value problems of order two. Atti.
Accad. Lincei 9 No 4 (1998) 245–266.
Ph. Clément, J. Prüss: An operator-valued transference principle and maximal regularity on vector-valued Lp -spaces. In: Evolution Equations and
Their Applications in Physics and Life Sciences. Lumer, Weis eds., Marcel
Dekker (2000) 67–87.
Ph. Clément, B. de Pagter, F.A. Sukochev, M. Witvliet: Schauder decomposition and multiplier theorems. Studia Math. 138 (2000) 135–163.
G. Da Prato, P. Grisvard: Sommes d’opérateurs linéaires et équations
différentielles opérationnelles. J. Math. Pure Appl. 54 (1975) 305–387.
G. Dore: Lp -regularity for abstract differential equations. In: Functional
analysis and related topics, editor: H. Komatsu. Springer LNM 1540 (1993)
25–38.
G. Dore, A. Venni: On closedness of the sum of two closed operators. Math.
Z. 196 (1987) 189–201.
R.E. Edwards and G.I. Gaudry: Littlewood-Paley and Multiplier Theory.
Springer, Berlin 1977.
N.J. Kalton, G. Lancien: A solution to the problem of the Lp -maximal regularity. Math. Z. (to appear).
S. Kwapien: Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients. Studia Math. 44 (1972) 583–595.
Y. Latushkin, R. Shvydkoy: Hyperbolicity of Semigroups and Fourier multipliers. Ulmer Seminare 2000.
H. Lelièvre: Espaces BM O, inégalités de Paley et multiplicateurs idempotents. Studia Math. 123 (1997) 249–274.
J. Lindenstrauss, L. Tzafriri: Classical Banach Spaces II. Springer, Berlin
1996.
A. Lunardi: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser, Basel 1995.
A. Mielke: Über maximale Lp -Regularität für Differentialgleichungen in
Banach- und Hilbert Räumen. Math. Ann. 277 (1987) 121–133.
J. van Neerven: The Asymptotic Behaviour of Semigroups of Linear Operators. Birkhäuser, Basel 1996.
G. Pisier: Les inégalités de Khintchine-Kahane d’après C. Borell. Séminaire
sur la géometrie des espaces de Banach 7, Ecole Polytechnique Paris 1977–
1978.
G. Pisier: Probabilistic methods in the geometry of Banach spaces. CIME
Summer School 1985, Springer Lecture Notes 1206 (1986) 167–241.
J. Prüss: On the spectrum of C0 -semigroups. Trans. Amer. Math. Soc. 284
(1984) 847–857.
S. Schweiker: Asymptotics, regularity and well-posedness of first- and
second-order differential equations on the line. Ph.D thesis, Ulm 2000.
Z. S̆trkalj and L. Weis: On operator-valued Fourier multiplier theorems.
Preprint 2000.
H. Tanabe: Equations of evolution Pitman, London, S.F., Melbourne 1979.
The operator-valued Marcinkiewicz multipler theorem
[W1]
[W2]
[Zi]
[Zy]
343
L. Weis: Operator-valued Fourier multiplier theorems and maximal Lp regularity. Preprint 2000.
L. Weis: A new approach to maximal Lp -regularity. Preprint 2000.
F. Zimmermann: On vector-valued Fourier multiplier theorems. Studia Math.
(1989) 201–222.
A. Zygmund: Trigonometric Series, Vol. II. Cambridge University Press
(1959).