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, Universit¨at 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 Classical theorems on Lp -multipliers are no longer valid for operator-valued functions unless the underlying space is isomorphic to a Hilbert space (see Sect. 1 for precise statements of this result).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 Cl´ement-de Pagter-Sukochev-Witvliet [CPSW], Weis [W1], [W2] and Cl´ement-Pr¨uss [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 Cl´ement-de Pagter-Sukochev and Witvliet [CPSW]). This research is part of the DFG-project: “Regularit¨at und Asymptotik f¨ur elliptische und parabolische Probleme”. The second author is supported by the Alexander-von-Humboldt Foundation and the NSF of China.
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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 Cl´ement-Pr¨uss [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 Straklj 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 H¨older 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 Pr¨uss [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.
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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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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 gˆ(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).
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˜ : T(X) → T(Y ) by M ˜ ( ek ⊗ xk ) = Proof. (i) ⇒ (ii). Define M ˜ has a unique extension M ∈ ek ⊗ Mk xk . Then by the assumption, M L(X, Y ). Then (1.3) follows by continuity. Clearly, (1.3) implies (1.2). (ii) ⇒ (i). Define M f = g with gˆ(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 Cl´ement-de Pagter-Sukochev-Witvliet [CPSW], Weis ˘ [W1], [W2], Strkalj-Weis [SW] and Cl´ement-Pr¨uss [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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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| 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 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
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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 )
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ˆ ˆ
≤
e ⊗ M + e ⊗ M f (k) f (k)
k k k k
k≥0
p k 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.
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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
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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.
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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
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sequence. Instead of assuming that k(Mk+1 − Mk ) is bounded, it suffices to assume that |Mk+1 − Mk | < ∞ sup j∈N
2j ≤|k| 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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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 Pr¨uss [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 Pr¨uss [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.
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4. H¨older continuous solution In this section we show that Pper has a unique H¨older 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→∞
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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
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
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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 u ˆ(k) = R(ik, A)fˆ(k) , vˆ(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 gˆ(k) = (ik)β w(k) ˆ 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
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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 vˆ(k) = −k 2 u ˆ(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.
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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 u ˆ(k) ∈ D(A) and 2 2 ˆ ˆ ˆ(k) + Aˆ u(k) = f (k). Hence u ˆ(k) = −R(k , A)f (k) and (u )ˆ(k) = −k u 2 2 2 ˆ(k) = k R(k , A)fˆ(k) (k ∈ Z). Since u ∈ Lp (0, 2π; X) this −k u 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 u ˆ(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 u ˆ(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) = gˆ(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 .
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Notice that f1 ∈ Lp (−π, π; X) exists as X is a U M D-space [Bur]. There ˆ 1 (k) (k ∈ Z). Since the exists h1 ∈ Lp (−π, π; X) such that Mk fˆ1 (k) = h p Riesz projection is bounded we find g1 ∈ L (−π, π; X) such that gˆ1 (k) = ˆ 1 (k) for k ≥ 0 and gˆ1 (k) = 0 for k < 0. Thus gˆ1 (k) = Mk fˆ(k) for k ≥ 0. h Similarly, we find g2 ∈ Lp (−π, π; X) such that gˆ2 (k) = Mk fˆ(k) for k < 0 and gˆ2 (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 u ˆ(k) + Aˆ u(k) = fˆ(k), hence u ˆ(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 u ˆ(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:
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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 Cl´ement and Guerre-Delabri`ere [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]
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