Arch. Math. 82 (2004) 404–414 0003–889X/04/050404–11 DOI 10.1007/s00013-003-0583-9 © Birkh¨auser Verlag, Basel, 2004

Archiv der Mathematik

Operator-valued Fourier multiplier theorems on Lp -spaces on Td By Shangquan Bu and Jin-Myong Kim

Abstract. We establish operator-valued Fourier multiplier theorems on Lp -spaces on Td . The conditions on the multipliers depend on the geometry of the underlying Banach spaces (UMD property and property (α)) and the growth rate (estimated by means of R-boundedness) at infinity of the partial derivatives of the multipliers. We also give an application of the obtained Fourier multiplier theorems to Lp -maximal regularity for a second order problem.

Let d ∈ N be fixed, T := [0, 2π]. In this paper the letter n will be reserved for the abbreviation of the vector (n1 , . . . , nd ) ∈ Zd . Let X be a complex Banach space and 1 < p < ∞. For f ∈ Lp (Td , X) and n ∈ Zd , we define the n-th Fourier coefficient of f by
dtd dt1 ... , f (t1 , . . . , td )e−n (t1 , . . . , td ) fˆ(n) := 2π 2π [0,2π]d

where en (t1 , . . . , td ) := ein1 t1 · · · eind td for ti ∈ T (1
i
d). f ∈ Lp (Td , X) is said to be an X-valued trigonometric polynomial on Td , if there exists N ∈ N such that f = fˆ(n)en with fˆ(n) ∈ X. We let T (Td , X) be the space of all X-valued trigonometric

|ni |
N

polynomials on Td . It is clear that the space T (Td , X) is dense in Lp (Td , X) (see [1, (1.1)] for a proof in the case d = 1). Let Y be another Banach space, we denote by L(X, Y ) the space of all bounded linear operators from X to Y . Let M = (M(n))n∈Zd ⊂ L(X, Y ). M is said to be a Fourier multiplier from Lp (Td , X) to Lp (Td , Y ), if the well defined linear operator from T (Td , X) to T (Td , Y ) given by (TM f )ˆ(n) := M(n)fˆ(n) (n ∈ Zd ) can be extended to a bounded linear operator from Lp (Td , X) to Lp (Td , Y ). When d = 1 and X, Y are UMD spaces, a sufficient condition on M ensuring that it defines a Fourier multiplier was obtained in [1, Theorem 1.3]: (M(n))n∈Z and Mathematics Subject Classification (2000): 42A45, 42B15, 46B20, 46E40. The first author is supported by the NSF of China and the Excellent Young Teacher Program of MOE, P. R. C.

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(n(M(n + 1) − M(n)))n∈Z (the derivative of M) are R-bounded. In this paper, we are interested to find sufficient conditions on M to ensure that it defines a Fourier multiplier when d > 1 and X, Y are UMD spaces. We will see that the conditions on M depend on the growth rate at infinity (estimated with the R-boundedness) of partial derivatives of M, they also depend on the geometry of the underlying Banach spaces, and in the proof of the results we use unconditional Schauder decompositions of Lp (Td , X) established in [11]. When X has the property (α), we have a finer unconditional Schauder decomposition of Lp (Td , X), it turns out that in this case the growth conditions at infinity of partial derivatives of M are weaker than in the general case (see Theorem 2 below). The concept of the R-boundedness of sets of operators is essential in the study of operatorvalued Fourier multipliers. Recall that a subset T ⊂ L(X, Y ) is Rademacher bounded (R-bounded, in short), if there exists C  0 such that      k  k           ( 0.1) γ T x
C γ x j j j j j   j =1 j =1   2

2

for all T1 , T2 , . . . , Tk ∈ T , x1 , x2 , . . . , xk ∈ X and k ∈ N [1] [2] [5] [9] [10], where γj is the j -th Rademacher function on [0, 1] given by γj (t) = sign(sin(2j π t)) [6]. We denote by R(T ) the smallest constant C  0 such that the inequality ( 0.1) holds true. By Kahane’s inequality [6, Theorem 1.e.13], we can replace the L2 -norm in the definition of R-boundedness by any other Lp -norm (1
p < ∞). It is easy to see from the definition that if T1 and T2 are R-bounded subsets of L(X, Y ) and L(Y, Z) respectively, the product T2 T1 := {ST : S ∈ T2 , T ∈ T1 } is R-bounded. If T ⊂ L(X, Y ) is n  R-bounded, then the complex absolute convex hull of T given by acon(T ) := { αi Ti : n ∈ N, Ti ∈ T , αi ∈ C,

n 

i=1

|αi |
1}, is still R-bounded and R(acon(T ))
2R(T )

i=1

[5, Lemma 3.2]. The same argument as in [1, Proposition 1.11] shows that if (M(n))n∈Zd is a Fourier multiplier from Lp (Td , X) to Lp (Td , Y ), then the set {M(n) : n ∈ Zd } must be R-bounded. We will use the notion of UMD spaces. Recall that a Banach space X is a UMD space, if there exists  (equivalently: if for all) 1 < p < ∞, suchpthat the Riesz projection defined fˆ(n)en is a bounded linear operator on L (T, X) [3] [4] [11]. It is known by Rf := n0

that if X is a UMD space, 1 < p < ∞ and if (, , µ) is any σ -finite measure space, the space Lp (µ, X) is still a UMD space. This implies in particular that if X is a UMD space, for any αi ∈ Z and any choice of signs i = ±1(1
i
d), there exists a constant C > 0 such that         ˆ f (n)en  ( 0.2)  
Cf p , i ni αi  p

whenever f ∈ Lp (T , X). This follows from the identification of Lp (Td , X) with Lp (T, Lp (Td−1 , X)) by Fubini’s Theorem. d

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Let M = (M(n))n∈Zd ⊂ L(X, Y ). For 1
j
d and n ∈ Zd , we let (Dj0 )M(n) := M(n), (Dj1 )M(n) := M(n + fj ) − M(n), (n ∈ Zd ) be the partial derivatives of M with respect to the j -th coordinate, where fj := (δj,h )1
h
d .  It is easy to verify that when 1
i, j
d and i , j ∈ {0, 1}, we have (Dii )(Dj j M)(n) = 

(Dj j )(Dii M)(n). Thus we can define the expression    d−1   Di i  M(n) := (Ddd Dd−1 · · · D11 )M(n) 1
i
d

without any confusion whenever i ∈ {0, 1} (1
i
d). The following result is one of the main results in this paper which generalizes

 the known result in the case d = 1 n2i for n ∈ Zd . [1, Theorem 1.3]. We use the notation |n| := 1
i
d

Theorem 1. Let X, Y be UMD spaces, d ∈ N, 1 < p < ∞ and let M ⊂ L(X, Y ) be an R-bounded subset. If (M(n))n∈Zd ⊂ L(X, Y ) satisfies       i   Dii  M(n) : n ∈ Zd , i ∈ {0, 1} (1
i
d) ⊂ M, |n| i    1
i
d

then (M(n))n∈Zd defines a Fourier multiplier from Lp (Td , X) to Lp (Td , Y ). Recall that an unconditional Schauder decomposition of a Banach space X is a family ∞  (Qk )k>0 of bounded linear projections on X such that Qk Q = 0 if k  = , and Qπ(k) k=1

x = x for all x ∈ X and for each permutation π : N → N. To state our next result, we need the subsequent geometric property of Banach spaces introduced by Pisier [7] and later used by Cl´ement-de Pagter-Sukochev-Witvliet [5] in the study of the interplay between R-boundedness and unconditional Schauder decompositions. A Banach space X has the property (α), if there exists a constant C  0 such that  1/2  2 
1
1   k      αij γi (t)γj (s)xij     dtds  i,j =1  0

0



1/2  2 
1
1   k     
C γi (t)γj (s)xij    dtds  i,j =1  0

0

for all xij ∈ X, αij = ±1 (i, j = 1, 2, . . . , k) and for all k ∈ N. In the proof of Theorem 1, we will use an unconditional Schauder decomposition of Lp (Td , X) when X is a UMD space. If X is a UMD space that has property (α), we

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have finer unconditional Schauder decompositions for Lp (Td , X). In this case, it turns out that we have a weaker sufficient condition on (M(n))n∈Zd ensuring that it defines a Fourier multiplier. When ni ∈ Z and i ∈ {0, 1} (1
i
d), we use the notation n := n11 · · · ndd . Note that |n |
|n|1 +···+d . Theorem 2. Let X and Y be UMD spaces that have property (α), 1 < p < ∞, d ∈ N. Let M ⊂ L(X, Y ) be an R-bounded subset. If (M(n))n∈Zd ⊂ L(X, Y ) satisfies        ( 0.3) Dii M(n) : n ∈ Zd , i ∈ {0, 1} (1
i
d) ⊂ M, n    1
i
d

then (M(n))n∈Zd defines a Fourier multiplier from Lp (Td , X) to Lp (Td , Y ). Moreover if we denote by S(M(n)) d the bounded linear operator from Lp (Td , X) to Lp (Td , Y ), then the set {S(M(n)) Lp (Td , Y )).

n∈Zd

n∈Z

: (M(n))n∈Zd satisfies condition ( 0.3)} is R-bounded in L(Lp (Td , X),

The key point in our proofs of Theorem 1 and 2 is the following result established in [5, Theorem 3.4 and Theorem 3.14]. Lemma 3. Let (Qn )n>0 be an unconditional Schauder decomposition of a Banach space X and let M be an R-bounded subset of L(X).  Assume that (Tn )n>0 ⊂ M is such that Tn Qn x converges in X for all x ∈ X, Tn Qn = Qn Tn for all n > 0. Then S(Tn )n>0 x := n>0

and S(Tn )n>0 is a bounded linear operator on X. Moreover if X has property (α), then the set {S(Tn )n>0 : (Tn )n>0 ⊂ M, Qn Tn = Tn Qn } is R-bounded in L(X). For fixed 1
j
d and k ∈ Z, if X is a UMD space, we let  (j ) Pk f := fˆ(n)en , nj
k, ni ∈ Z (i=j )

(j )

for f ∈ Lp (Td , X). The operator Pk is bounded on Lp (Td , X) as Lp (Td−1 , X) is a UMD space. The proofs of Theorem 1 and 2 also use the following lemma. (j )

Lemma 4. If X is a UMD space, then the sequence (Pk )k∈Z is R-bounded. P r o o f. For k ∈ Z, let φk,j be the bounded linear operator on Lp (Td , X) defined by (φk,j f )(t1 , . . . , td ) := e−iktj f (t1 , . . . , td ), ((t1 , . . . , td ) ∈ Td ). (j )

It is easy to verify that Pk [5, Corollary 3.18].


(j )

= φ−k,j P0 φk,j when k ∈ Z. Hence the result follows from

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P r o o f o f T h e o r e m 1. We first consider the case X = Y . Assume that X is a UMD space, 1 < p < ∞. Let Z = {f ∈ Lp (Td , X) : fˆ(n) = 0 if there exists 1
i
d such that ni
1}. It is clear that Z is a closed subspace of Lp (Td , X). By ( 0.2), in order to show that (M(n))n∈Zd is a Fourier multiplier on Lp (Td , X), it suffices to show that there exists C > 0 such that        ˆ(n)en 
Cf p ( 0.4) M(n) f    ni >1 p

for all X-valued trigonometric polynomials f ∈ Z. Let k ∈ N, k  d + 1. There exist unique r  1 and 1
j
d, such that k = dr + j . We let Dk = {n ∈ Zd : 1
ni
2r+1 − 1(1
i
j − 1), 1
ni
2r − 1 (j + 1
i
d), 2r
nj
2r+1 − 1}, and Qk f =



fˆ(n)en , (f ∈ Lp (Td , X)).

n∈Dk

By [11, Proposition 1], the Schauder decomposition (Qk )k>d of Z is unconditional. Let f ∈ Z be an X-valued trigonometric polynomial on Td , we have      M(n)fˆ(n)en = Qk  M(n)fˆ(n)en  ni >1

=

 k>d

=

 k>d

=



 

k>d



n∈Dk

  



n∈Dk

 n∈Dk

k>d

ni >1



(d) (1) (Pn(d) − Pnd −1 ) · · · (Pn(1) − Pn1 −1 )M(n)Qk f d 1









(−1) i

 i

(d) (1) Pnd −d · · · Pn1 −1  M(n)Qk f

i ∈{0,1}









(−1) i



 i

Mk (n1 + 1 , . . . , nd + d )

i ∈{0,1}



Pn(d) · · · Pn(1) Qk f d 1 =

 k>d

 



n∈Dk

 (−1)d 

 1
i
d





Qk f, Di1  Mk (n)Pn(d) · · · Pn(1) d 1

where Mk (n) = M(n) if n ∈ Dk , and Mk (n) = 0 if n ∈ / Dk , and Dk = {n ∈ Zd : 0
ni
2r+1 − 1 (1
i
j − 1), 0
ni
2r − 1 (j + 1
i
d), 2r − 1
nj
2r+1 − 1}.

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To show ( 0.4), by Lemma 3 it suffices to show that        (1)  R ( 0.5) Di1  Mk (n)Pn(d) · · · P : k > d < ∞. n d 1    n∈Dk

1
i
d

Let αi = 0, βi = 2r+1 − 1 for 1
i
j − 1, αi = 0, βi = 2r − 1 for j + 1
i
d, and αj = 2r − 1, βj = 2r+1 − 1. For S, T ⊂ {1, 2, . . . , d}, S ∩ T = ∅, we let Dk,S,T = {n ∈ Dk : ni = αi (i ∈ S), ni = βi (i ∈ T ), αi < ni < βi (i ∈ / S ∪ T )}. Then Dk = ∪S,T ⊂{1,2,...,d},S∩T =∅ Dk,S,T . For n ∈ Dk,S,T , we have          1 |T | 1  Di  Mk (n) = (−1) Di M n + fl , i ∈S∪T /

1
i
d

l∈S

where |T | denotes the number of elements in T . We deduce that          R Di1  Mk (n)Pn(d) · · · Pn(1) : k>d d 1   n∈Dk,S,T 1
i
d       |T | 1 = R (−1) Di  n∈Dk,S,T i ∈S∪T /     M n+ fl Pn(d) · · · Pn(1) :k>d d 1 l∈S

  

= R

|T |



1

d−|S∪T |

(−1) |n|  |n|d−|S∪T | n∈Dk,S,T     (d) (1) M n+ fl Pnd · · · Pn1 : k > d l∈S


sup

k>d n∈D

k,S,T

1 |n|d−|S∪T |

 n∈Dk,S,T



 R(acon(M))

We have R(acon(M)) < ∞ [5, Lemma 3.2] and We remark that when n of the set Dk,S,T is not

i ∈S∪T /





1




 Di1

(i)

R{Pl

: l ∈ Z}.

1
i
d



(i)

R{Pl

1
i
d ∈ Dk,S,T , we have |n|  2r −1  2r−1 greater than 2(r+1)(d−|S∪T |) . Hence

|n|d−|S∪T |



: l ∈ Z} < ∞ by Lemma 4. and the number of elements

2(r+1)(d−|S∪T |)
4d . 2(r−1)(d−|S∪T |)

This implies that ( 0.5) holds true and thus finishes the proof in the case X = Y .

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Now we consider the general case. Since X and Y are UMD-spaces , X ⊕ Y is a ˜ ˜ UMD-space also. Define M(n) ∈ L(X ⊕ Y ) by M(n)(x, y) = (0, M(n)x). It follows from ˜ the first case that (M(n))n∈Zd is a Fourier multiplier on Lp (Td , X ⊕ Y ). It is easy to see that this implies that (M(n))n∈Zd is a Fourier multiplier from Lp (Td , X) to Lp (Td , Y ). P r o o f o f T h e o r e m 2. We first assume that X = Y and let Z = {f ∈ Lp (Td , X) : fˆ(n) = 0 if there exists 1
i
d such that ni
0}. It is clear that Z is a closed subspace of Lp (Td , X). Let N0 := N ∪ {0}. For µ = (µ1 , . . . , µd ) ∈ Nd0 , we let Dµ = {n ∈ Zd : 2µi
ni
2µi +1 − 1 (1
i
d)}. Since X is a UMD space and has property (α), if we define  Qµ f = fˆ(n)en , (f ∈ Lp (Td , X)), n∈Dµ

then the family (Qµ )µ∈Nd is an unconditional Schauder decomposition of Z 0 [11, Proposition 1] (Zimmermann’s result was stated under the assumption that X has l.u.st., but the result remains valid with the weaker assumption that X has property (α)). By ( 0.2), in order to show the theorem we only need to consider the restriction to Z. If f ∈ Z is an X-valued trigonometric polynomial, we have by the same computation used in the proof of Theorem 1 that  M(n)fˆ(n)en ni >0

=

 µ

 



 (−1)d 

 n∈Dµ



1
i
d





 Qµ f, Di1  Mµ (n)Pn(d) · · · Pn(1) d 1

where Dµ = {n ∈ Zd : 2µi − 1
ni
2µi +1 − 1 (1
i
d)}, / Dµ . and Mµ (n) = M(n) if n ∈ Dµ , Mµ (n) = 0 if n ∈ Let αi = 2µi − 1, βi = 2µi +1 − 1 for 1
i
d. For S, T ⊂ {1, 2, . . . , d}, S ∩ T = ∅, we let Dµ,S,T = {n ∈ Dµ : ni = αi (i ∈ S), ni = βi (i ∈ T ), αi < ni < βi (i ∈ / S ∪ T )}. Then Dµ = ∪S,T ⊂{1,2,...,d},S ∩ T =∅ Dµ,S,T . For n ∈ Dµ,S,T ,  



1
i
d

 Di1  Mµ (n)

 |T |

= (−1)



i ∈S∪T /

 Di1

 M n+

 l∈S

 fl .

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By Lemma 3, it suffices to show that there exists an R-bounded subset N ⊂ L(Lp (Td , X), Lp (Td , Y )) such that      (−1)|T | Di1 n∈Dµ,S,T



M n+

( 0.6)



 fl

i ∈S∪T /



· · · Pn(1) Pn(d) d 1

:µ∈

Nd0

⊂ N.

l∈S (i)

We know by Lemma 4 that the sets {Pni : ni ∈ Z} (1
i
d) are R-bounded. Since the product of R-bounded subsets is still R-bounded and the complex absolute convex hull of an R-bounded subset is R-bounded, to show ( 0.6), by the assumption it will suffice to show that   sup ( 0.7) n−1 i < ∞. µ

n∈Dµ,S,T i ∈S∪T /

We have for µ ∈ Nd0 ,   n∈Dµ,S,T i ∈S∪T /

n−1 i =





αi
n−1 i =





αi
n−1 i
1.

This shows that ( 0.7) holds true and thus finishes the proof in the case X = Y . The proof for the general case is exactly the same as in the proof of Theorem 1. R e m a r k 5. In [8], operator-valued Fourier multiplier theorems on Lp (Td , X) were also obtained. Actually our Theorem 1 and the first part of Theorem 2 can be deduced from the results in [8], but the proofs given there are more complicated and our conditions on the multipliers M are more readable and easier to check in the practice. Note that the second part of our Theorem 2 is new. Also in [8], they use the obtained Fourier multiplier theorems on Lp (Td , X) to get Fourier multiplier theorems on Lp (Rd , X), actually using our results we can also deduce the same operator-valued Fourier multiplier theorems on Lp (Rd , X) and the proofs are simpler. In the last part of this paper, we give an application of Theorem 1. Let X be a complex 1,p Banach space and 1
p < ∞. We let Hper (T, X) := {u ∈ Lp (T, X) : ∃v ∈ Lp (T, X), 1,p v(k) ˆ = ik u(k)(k ˆ ∈ Z)} be the first Sobolev space [1]. By [1, Lemma 2.1] u ∈ Hper (T, X) 2π  if and only if there exist x ∈ X and v ∈ Lp (T, X), such that v(s)ds = 0 and u(t) = x+

t

0

v(s)ds a.e. on T. This implies that each u ∈

1,p Hper (T, X)

has a unique continuous

0 2,p

representative and this representative is a.e. differentiable. We let Hper (T, X) := {u ∈ 1,p 1,p Hper (T, X) : u ∈ Hper (T, X)} be the second Sobolev space. Let u ∈ Lp (T, X),

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2,p

it is easy to see that u ∈ Hper (T, X) if and only if there exists v ∈ Lp (T, X) such that 2,p v(k) ˆ = −k 2 u(k) ˆ (k ∈ Z) [1]. Each element in Hper (T, X) has a continuously differentiable representative u, and u is a.e. differentiable. To introduce the Laplacian on Lp (Td , X), we let Ti be a copy of T (1
i
d). By   Ti , X) (= Lp (Td , X)) and Lp (Tj , Lp ( Ti , X)) Fubini’s Theorem, the spaces Lp ( i=j 1
 i
d Ti , X), we denote the corresponding element in can be identified. For u ∈ Lp ( 1
i
d  Lp (Tj , Lp ( Ti , X)) by Tj u. We define the second Sobolev space by letting i=j

 2,p Hper







Ti , X 

1
i
d

           2,p := u ∈ Lp  Ti , X  : Tj u ∈ Hper Tj , Lp  Ti , X  (1
j
d) .   i=j

1
i
d

Let (M(n))n∈Zd ⊂ L(X) and 1
j
d, for fixed n j ∈ Z, we consider the sequence N(nj )j ∈Z of bounded linear operators on Lp ( Ti , X) given by the Fourier i=j

multiplier (M(n))ni ∈Z (i=j ) . It is not hard to see that (M(n))n∈Zd defines a Fourier  multiplier on Lp ( Ti , X) if and only if N (nj )j ∈Z defines a Fourier multiplier on 1
i
d

2,p  Ti , X)). This implies in particular that u ∈ Hper ( Ti , X) if and only i=j 1
i
d  if for 1
j
d, there exists vj ∈ Lp ( Ti , X) such that vˆj (n) = −n2j u(n) ˆ (n ∈ Zd ).

Lp (Tj , Lp (



2,p

For u ∈ Hper (



1
i
d



1
i
d

Ti , X), we define the Laplacian u by u := 2,p



1
j
d

(Tj u) ∈

Ti , X) and u, it is no longer  necessary to distinguish Ti for 1
i
d. Thus we will identify Ti with Td . Lp

1
i
d

Ti , X . After the definition of Hper (



1
i
d

1
i
d

Let A : D(A) → X be a closed linear operator in X, we consider the problem ( 0.8)

u(t) + A(u(t)) = f (t) a.e. on Td ,

where f ∈ Lp (Td , X) is given. We say that the problem ( 0.8) has Lp -maximal regularity, if 2,p for each f ∈ Lp (Td , X), there exists a unique u ∈ Hper (Td , X) ∩ Lp (Td , D(A)) satisfying the equation ( 0.8), where we consider D(A) as a complex Banach space equipped with its graph norm. We denote the resolvent (λ − A)−1 by R(λ, A) when λ ∈ ρ(A). We have the following consequence of Theorem 1. Theorem 6. Let X be a UMD space, d ∈ N and let 1 < p < ∞. Then the following assertions are equivalent:

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Operator-valued Fourier multiplier theorems on Lp -spaces on opens Td

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(i). The problem ( 0.8) has Lp -maximal regularity, (ii). {|n|2 : n ∈ Zd } ⊂ ρ(A) and (|n|2 R(|n|2 , A))n∈Zd is R-bounded, (iii). {|n|2 : n ∈ Zd } ⊂ ρ(A) and (|n|2 R(|n|2 , A))n∈Zd is a Fourier multiplier on Lp (Td , X). P r o o f. It is clear that (iii) implies (ii). The converse implication follows from the argument used in the proof of [1, Theorem 2.3] and Theorem 1 (one shows that the sequence in (iii) satisfies the assumption in Theorem 1). We only need to show the equivalence between (i) and (iii). Assume that the Problem ( 0.8) has Lp -maximal regularity, one shows similarly as in [1, Theorem 2.3] that {|n|2 : n ∈ Zd } ⊂ ρ(A). Let f ∈ Lp (Td , X) and let u ∈ 2,p Hper (Td , X) ∩ Lp (Td , D(A)) be the unique solution of ( 0.8). Then by [1, Lemma 3.1], we have u(n) ˆ ∈ D(A) and −|n|2 u(n) ˆ + Au(n) ˆ = fˆ(n) (n ∈ Zd ). Thus u(n) ˆ = − d d 2 R(|n| , A)fˆ(n) (n ∈ Z ). We have u ∈ Lp (T , X) by assumption, this implies that ( u)ˆ(n) = |n|2 R(|n|2 , A)fˆ(n) (n ∈ Zd ). Hence we have shown (iii). Conversely, assume that {|n|2 : n ∈ Zd } ⊂ ρ(A) and (|n|2 R(|n|2 , A))n∈Zd defines a Fourier multiplier on Lp (Td , X). Since (Nn )n∈Zd defined by Nn = |n|−2 if n  = 0, and N0 = 0 satisfies the assumption of Theorem 1, the sequence (R(|n|2 , A))n∈Zd also defines a Fourier multiplier on Lp (Td , X). It follows from the equality AR(|n|2 , A) = |n|2 R(|n|2 , A) − I that (R(|n|2 , A))n∈Zd is a Fourier multiplier from Lp (Td , X) to Lp (Td , D(A)). For f ∈ Lp (Td , X), there exist v ∈ Lp (Td , D(A)) and u ∈ Lp (Td , X) ˆ = |n|2 R(|n|2 , A)fˆ(n) (n ∈ Zd ). We claim that v ∈ such that v(n) ˆ = R(|n|2 , A)fˆ(n), u(n) 2,p d Hper (T , X). For this it suffices to show that for 1
j
d, (n2j R(|n|2 , A))n∈Zd is a Fourier multiplier on Lp (Td , X). This follows from the hypothesis that (|n|2 R(|n|2 , A))n∈Zd defines a Fourier multiplier on Lp (Td , X), and the fact that (n2j |n|−2 I )n∈Zd also defines a Fourier multiplier on Lp (Td , X) (one verifies that it satisfies the assumption in Theorem 1). ˆ + Av(n) ˆ = fˆ(n) (n ∈ Zd ), we deduce By [1, Lemma 3.1] and the equality −|n|2 v(n) that v satisfies ( 0.8) and we thus prove the existence. The uniqueness follows from the same argument used in the proof of [1, Theorem 2.3].
References [1] W. Arendt and S. Bu, The operator-valued Marcinkiewicz multiplier theorem and maximal regularity. Math. Z. 240, 311–343 (2002). [2] W. Arendt and S. Bu, Tools for maximal regularity. Math. Proc Cambridge Philos. Soc. 134, 317–336 (2003). [3] J. Bourgain, Vector valued singular integrals and the H 1 − BMO duality. In: Probability Theory and Harmonic Analysis, Burkholder ed., 1–19, New York 1986. [4] D. Burkholder, A geometric condition that implies the existence of certain singular integrals of Banachspace-valued functions. In: Proc. of Conf. on Harmonic Analysis in Honor of Antoni Zygmund, Chicago 1981, 270–286, Belmont, Calif. 1983. [5] Ph. Cl´ement, B. de Pagter, F. A. Sukochev and M. Witvliet, Schauder decomposition and multiplier theorems. Studia Math. 138, 135–163 (2000). [6] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II. Berlin 1996.

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Shangquan Bu and Jin-Myong Kim

arch. math.

[7] G. Pisier, Some results on Banach spaces without local unconditional structure. Compositio Math. 37, 3–19 (1978). ˘ [8] Z. Strkalj and L. Weis, On operator-valued Fourier multiplier theorems. Preprint 2000. [9] L. Weis, Operator-valued Fourier multiplier theorems and maximal Lp -regularity. Math. Ann. 319, 735–758 (2001). [10] L. Weis, A new approach to maximal Lp -regularity. In: Evolution Equations and Their Applications in Physics and Life Sciences. Lumer and Weis, eds., 195–214, New York 2000. [11] F. Zimmermann, On vector-valued Fourier multiplier theorems. Studia Math. 93, 201–222 (1989). Received: 10 January 2003; revised manuscript accecpted: 17 February 2003 Shangquan Bu Department of Mathematical Science University of Tsinghua Beijing 100084 China sbu@math.tsinghua.edu.cn

Jin-Myong Kim Department of Mathematics University of Kim Il Sung DPR Korea Current Address Department of Mathematical Science University of Tsinghua Beijing 100084 China jjin@math.tsinghua.edu.cn