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1 L p MAXIMAL REGULARITY ON BANACH SPACES WITH A SCHAUDER BASIS N. J. KALTON AND G. LANCIEN Abstract. We investigate the problem of L p -maximal regularity on Banach spaces having a Schauder basis. Our results improve those of a recent paper. We also address the question of L r -regularity in L s spaces. 1. Introduction We will only recall the basic facts and definitions on maximal regularity. For further information, we refer the reader to [], [4], [8] or [7]. We consider the following Cauchy problem: { u (t) + B(u(t)) = f(t) for t < T u() = where T (, + ), B is the infinitesimal generator of a bounded analytic semigroup on a complex Banach space X and u and f are X-valued functions on [, T ). Suppose 1 < p <. B is said to satisfy L p maximal regularity if whenever f L p ([, T ); X) then the solution u(t) = t e (t s)b f(s) ds satisfies u L p ([, T ); X). It is known that B has L p -maximal regularity for some 1 < p < if and only if it has L p -maximal regularity for every 1 < p < [3], [4], [14]. We thus say simply that B satisfies maximal regularity (MR). As in [7], we define: Definition 1.1. A complex Banach space X has the maximal regularity property (MRP) if B satisfies (MR) whenever B is the generator of a bounded analytic semigroup. Let us recall that De Simon [3] proved that any Hilbert space has (MRP), and that the question whether L q for 1 < q < has (MRP) remained open until recently. Indeed, in [7] it is shown that a Banach space with an 1991 Mathematics Subject Classification. Primary: 47D6, Secondary: 46B3, 46B5, 46C15. The first author was partially supported by NSF grant DMS

2 N. J. KALTON AND G. LANCIEN unconditional basis (or more generally a separable Banach lattice) has (MRP) if and only if it is isomorphic to a Hilbert space. In this paper we attempt to work without these unconditionality assumptions and study the (MRP) on Banach spaces with a finite-dimensional Schauder decomposition. In particular, we show that a UMD Banach space with an (FDD) and satisfying (MRP) must be isomorphic to an l sum of finite dimensional spaces. In the last question we consider the question of whether the solution u of our Cauchy problem satisfies u L ([, T ; L r ) if f L ([, T ); L s ). This work was done during a visit of the second author to the Department of Mathematics of the University of Missouri in Columbia in fall 1999; he would like to thank the Department for its warm hospitality.. Notation and background We will follow the notation of [7]. Let us now introduce more precisely a few notions. If F is a subset of the Banach space X, we denote by [F ] the closed linear span of F. We denote by (ε k ) k= the standard sequence of Rademacher functions on [, 1] and by (h k ) k= the standard Haar functions on [, 1] (for convenience we index from ). Let 1 p <. A Banach space X has type p if there is a constant C > such that for every finite sequence (x k ) K in X: ( 1 K K ε k (t)x k dt) 1/ C( x k p ) 1/p. Notice that every Banach space is of type 1. A Banach space X is called (UMD) if martingale difference sequences in L ([, 1]; X) are unconditional i.e. there is a constant K so that for every martingale difference sequence (f n ) N we have δ k f k L (X) K f k L (X) if sup k N δ k 1. Let (E n ) n 1 be a sequence of closed subspaces of X. Assume that (E n ) n 1 is a Schauder decomposition of X and let (P n ) n 1 be the associated sequence of projections from X onto E n. For convenience we will also denote this Schauder decomposition by (E n, P n ) n 1. The decomposition constant is defined by sup n n K=1 P k; this is necessarily finite. If each (E n ) is finitedimensional we refer to (E n ) as an (FDD) (finite-dimensional decomposition); an unconditional (FDD) is abbreviated to (UFDD).

3 L p MAXIMAL REGULARITY ON BANACH SPACES WITH A SCHAUDER BASIS 3 If (E n ) n 1 is a Schauder decomposition of X and (u n ) N is a finite or infinite sequence (i.e. N ) of the form u n = r n k=r n 1 +1 x k where x k E k and 1 = r < r 1 <.. < r n <.., then (u n ) n 1 is called a block basic sequence of the decomposition (E n ). We denote by ω <ω the set of all finite sequences of positive integers, including the empty sequence denoted. For a = (a 1,.., a n ) ω <ω, a = n is the length of a ( = ). For a = (a 1,.., a k ) (respectively a = ), we denote (a, n) = (a 1,.., a k, n) (respectively (a, n) = (n)). A subset β of ω <ω is a branch of ω <ω if there exists (σ n ) N such that β = {(σ 1,.., σ n ); n 1}. In this paper, for a Banach space X, we call a tree in X any family (y a ) a ω <ω X. A tree (y a ) a ω <ω is weakly null if for any a ω <ω, (y (a,n) ) n 1 is a weakly null sequence. Let (y a ) a ω <ω be a tree in the Banach space X. Let T ω <ω, (y a ) a T is a full subtree of (y a ) a ω <ω if T and for all a T, there are infinitely many n N such that (a, n) T. Notice that if (y a ) a T is a full subtree of a weakly null tree (y a ) a ω <ω, then it can be reindexed as a weakly null tree (z a ) a ω <ω We now state a result of [7] that will be an essential tool for this paper: Theorem.1. Let (E n, P n ) n 1 be a Schauder decomposition of the Banach space X. Let Z n = PnX and Z = [ Z n ]. Assume X has (MRP). Then there is a constant C > so that whenever (u n ) N are such that u n [E n 1, E n ] and (u n) N are such that u n [Z n 1, Z n ] then ( ) 1/ ( π ) 1/ P n u n e int dt π C u n e int dt π π and ( π ) 1/ ( Pnu ne int dt π C π ) 1/ u ne int dt. π We observe that, by a well-known result of Pisier [1] these inequalities can be replaced by equivalent inequalities (with a modified constant) using ε k in place of e ikt : (.1) P n u n ε n L (X) C u n ε n L (X) and (.) P n u nε n L (X) C u nε n L (X ). We refer the reader to [15] for further recent developments in this area.

4 4 N. J. KALTON AND G. LANCIEN 3. The main results We begin with a general result on spaces with a Schauder decomposition: Theorem 3.1. Let X be a Banach space of type p > 1 and with a Schauder decomposition (E n ). If X has (MRP), then there is a constant C > so that for any block basic sequence (u k ) N with respect to the decomposition (E n ): (3.3) 1 C u k 1 ε k (t)u k dt C K u k. Proof. If the result is false we can clearly inductively construct an infinite normalized block basic sequence (u n ) so that there is no constant C so that for all finitely nonzero sequences (a k ) we have: (3.4) 1 C a k 1 a k ε k (t)u k dt C a k It therefore suffices to show that (3.4) holds for every normalized block basic sequence (u n ). We can clearly then suppose u n E n. We next use a theorem of Figiel and Tomczak-Jaegermann [5] combined with [13] (see also [1] p.11) that, since X has nontrivial type for every n N there exists ϕ(n) N so that any subspace F of X with dimension ϕ(n) has a subspace H of dimension n which is -complemented in X and -isomorphic to l n. Assume (3.4) is false. Then we can inductively find a sequence (a n ) n 1 and an increasing sequence (r n ) n with r = so that r n > r n 1 +ϕ(r n 1 r n ) for n 1, and either or 1 1 n+1 n+1 r n +1 k=r n +1 n+1 k=r n +1 a k = 1 a k ε k (t)u k dt > n a k ε k (t)u k dt < n. In order to create new Schauder decompositions of X, we will need the following elementary lemma, that we state without a proof:

5 L p MAXIMAL REGULARITY ON BANACH SPACES WITH A SCHAUDER BASIS 5 Lemma 3.. Let (E n ) n 1 be a Schauder decomposition of a Banach space X. Assume that each E n has a finite Schauder decomposition (F n,k ) mn with a uniform bound on the decomposition constant. Then (F 1,1,, F 1,m1, F,1,, F,m, ) is also a Schauder decomposition of X. We denote the induced decomposition by ( m n F n,k). Now by assumption E rn E rn which has dimension at least ϕ(r n r n 1 ) contains a subspace H n which is -Hilbertian and -complemented in X. Let G n be the complement of H n in E rn E rn by the projection of norm. At the same time [u k ] is 1-complemented (by the Hahn-Banach theorem) in E k for r n 1 +1 k r n and let F k be its associated complement. We thus have a new Schauder decomposition: (F 1, [u 1 ], F, [u ],, F r1, [u r1 ], H 1, G 1, F r +1, [u r +1],, [u r3 ], H, G, ). If we write D n = F rn F rn 1 + G n then we have a Schauder decomposition r n (D n H n [u k ]). k=r n +1 Next select a normalized basis (v k ) r n 1 k=r n +1 of H k which is -equivalent to the canoncal basis of l r n r n 1. It is easy to see that we can obtain a new Schauder decomposition by interlacing the (v k ) with the (u k ) i.e.: (3.5) (D n [u rn +1] [v rn +1] [u rn 1 ] [v rn 1 ]). Now again using Lemma 3. we can form two further decompostions: (3.6) (D n [u rn +1+v rn +1] [v rn +1] [u rn 1 +v rn 1 ] [v rn 1 ]), and (3.7) (D n [u rn +1+v rn +1] [u rn +1] [u rn 1 +v rn 1 ] [u rn 1 ]). Now we can apply Theorem.1. If we use decomposition (3.6) we note that u k = (u k + v k ) v k and so for a suitable C and all n, r n k=r n +1 a k (u k + v k )ε k L (X) C n 1 k=r n +1 a k u k ε k L (X). However, using decomposition (3.5) there is also a constant C so that r n k=r n +1 a k v k ε k L (X) C r n k=r n +1 a k (u k + v k )ε k L (X).

6 6 N. J. KALTON AND G. LANCIEN This leads to an estimate: n 1 k=r n +1 a k 1 C 1 n 1 k=r n +1 a k u k ε k L (X). If we use decomposition (3.7) instead we obtain an estimate: 1 n 1 k=r n +1 a k u k e ikt L (X) C n 1 k=r n +1 Combining gives us (3.4) and completes the proof. a k Let us first use this result to give a mild improvement of a result from [7]: Theorem 3.3. Let X be a reflexive space with an (FDD) and with non-trivial type which embeds into a space Y with a (UFDD). If X has (MRP) then X is isomorphic to an l sum of finite-dimensional spaces ( E n) l. Proof. Using Proposition 1.g.4 of [9] (cf. [6]) we can block the given (FDD) to produce an (FDD) (E n ) so that (E n ) and (E n 1 ) are both (UFDD) s. Let us denote, as in Theorem.1, the dual (FDD) of X by (Z n ). Now it follows applying Theorem 3.1 to both X and X (which also has (MRP)) that there exists a constant C so that if x n E n and x n Z n are two finitely nonzero sequences x k j C( x k j ) 1 for j =, 1. Hence x k j C( x k j ) 1 x k C( x k ) 1 x k C( x k ) 1. Now for given x k we may find yk X with yk = x k and y k (x k ) = x k. Let x k = P k y k (where P k : X E k is the projection associated with the FDD (E n )). Then x k C 1x k where C 1 = sup n P n <. Hence if (x k ) is finitely nonzero, we have x k CC 1 ( x k ) 1..

7 Thus L p MAXIMAL REGULARITY ON BANACH SPACES WITH A SCHAUDER BASIS 7 x k = x k(x k ) = ( x k)( x k ) CC 1 ( x k ) 1 x k so that we obtain the lower estimate: ( x k ) 1 CC1 x k. This completes the proof. We next give another application to (UMD)-spaces with (MRP). Theorem 3.4. Let X be a (UMD) Banach space with an (FDD) satisfying (MRP). Then X is isomorphic to an l -sum of finite dimensional spaces, ( E n) l. Proof. Let (E n ) be the given (FDD) of X. We will show first that there is a blocking (F n ) of (E n ) which satisfies an upper estimate i.e. if there is a constant A so that if (x n ) is block basic with respect to (F n ) and finitely non-zero then (3.8) x n A( x n ) 1. Once this is done, the proof can be completed easily. Indeed if (Z n ) is the dual decomposition to (F n ) for X then we can apply the fact that X also has (MRP) (X is reflexive) to block (Z n ) to obtain a decomposition which also has an upper -estimate. Thus we can assume (F n ) and (Z n ) both have an upper -estimate and then repeat the argument used in Theorem 3.3 to deduce that X = ( F n) l. Since X necessarily has type p > 1, we can apply Theorem 3.1 and assume (E n ) obeys (3.3). We now introduce a particular type of tree in the space L ([, 1); X). Let D n for n be the sub-algebra of the Borel sets of [, 1) generated by the dyadic intervals [(k 1) n, k n ) for 1 k n. Let E n denote the conditional expectation operator E n f = E(f D n ). We will say that a tree (f a ) a ω <ω is a martingale difference tree or (MDT) if each f a is D a measurable, if a > then E a 1 f a =,

8 8 N. J. KALTON AND G. LANCIEN there exists N so that if a > N then f a =. In such a tree the partial sums along any branch form a dyadic martingale which is eventually constant. We will prove the following Lemma: Lemma 3.5. There is a constant K so that if (f a ) a ω <ω is a weakly null (MDT), there is a full subtree (f a ) a T so that for any branch β we have: f a L (X) K( f a L (X)) 1. Proof. For each a we define integers m (a) and m + (a). If f a we set m (a) to be the greatest m so that m P m f a L (X) a 1 f a L (X) and m + (a) to be the least m > m (a) so that P k f a L (X) a 1 f a L (X). k=m+1 If f = we set m ( ) = and m + ( ) = 1; if f a = where a we set m (a) to be the last member of a and m + (a) = m (a) + 1. Since (f a ) is weakly null we have lim n m (a, n) = for every a. It is then easy to pick a full subtree T so that m (a, n) > m + (a) whenever a, (a, n) T. Now let g a = m + (a) k=m (a)+1 f a. Then f a g a L (X) a f a L (X). For any branch β of T, we have that g a (t) is a block basic sequence with respect to (E n ) for every t < 1. Hence ( 1 ) 1 ( ɛ a (s)g a (t) Xds C g a (t) X Integrating again we have ( 1 From this we get ( 1 ɛ a (s)f a L (X)ds ɛ a (s)g a L (X)ds ) 1 ) 1 ( C g a L (X) ( C f a L (X) ) 1 ) 1. ) 1 +. a f a. Estimating the last term by the Cauchy-Schwarz inequality and using the fact that X is (UMD) we get the Lemma.

9 L p MAXIMAL REGULARITY ON BANACH SPACES WITH A SCHAUDER BASIS 9 Now we introduce a functional Φ on X by defining Φ(x) to be the infimum of all λ > so that for every weakly null (MDT) (f a ) a ω <ω with f = xχ [,1) we have a full subtree T so that for any branch β (3.9) Note that that since f a L (X) λ + K a f a L (X) (x + a f a L (X). f a L (X)) we have an estimate Φ(x) x. By considering the null tree we have F (x) x. It is clear that Φ is continuous and -homogeneous. Most importantly we observe that Φ is convex; the proof of this is quite elementary and we omit it. It follows that we can define an equivalent norm by x = Φ(x) and x x x for x X. Next we prove that if x X and (y n ) is a weakly null sequence then (3.1) lim sup( x + y n + x y n ) x + 4K lim sup y n. n n We first note that we can suppose lim n x ± y n and lim n y n all exist. Now suppose ɛ >. Then we can find weakly null (MDT) s (fa n ) a ω <ω with f n x+y n so that for every full subtree T we have a branch β on which: (3.11) f n a L (X) + ɛ > x + y n + K a f n a L (X). In fact by easy induction we can pick a full subtree so that (3.11) holds for every branch. Hence we suppose the original tree satisfies (3.11) for every branch. Similarly we may find weakly null (MDT) s (ga n ) a ω <ω with g n x y n and for every branch β, g n a L (X) + ɛ > x y n + K a We next consider the (MDT) defined by h x, { y n if t < 1 h (n) (t) = y n if 1 t < 1 and if a > 1 then h (a,n) (t) = { fa n (t 1) if t < 1 ga n (t) if 1 t < 1. g n a L (X).

10 1 N. J. KALTON AND G. LANCIEN Now for every branch of the (MDT) (h a ) a ω <ω have with initial element {n} we h a L (X) + ɛ > 1 ( x + y n + x y n ) + K a >1 h a L (X). However, from the definition of Φ(x) = x it follows that there exists n so that if n n we can find a branch β whose initial element is n and such that h a L (X) < x + K h a L (X) + ɛ. a > Combining gives the equation (for n n ), 1 ( x + y n + x y n ) x + K y n + ɛ. This proves (3.1). But note that if y n is weakly null we have lim inf n x y n x and so we deduce: lim sup x + y n x + 4K lim sup y n. n Using this equation it is now easy to block the Schauder decomposition (E n ) to produce a Schauder decomposition (F n ) with the property that for any N if x F F N and y k=n+ F k then x + y (1 + δ N )( x + 4K y ) 1, where δ N > are chosen to be decreasing and so that N=1 (1 + δ N). Next suppose (x k ) is any finitely non-zero block basic sequence with respect to (F n ). By an easy induction we obtain for j =, 1: n 1 x k j 4K (1 + δ k j )( x k j ) 1. Hence x k 3K ( x k ) 1. This establishes (3.8) and as shown earlier this suffices to complete the proof. Remark. Recently Odell and Schlumprecht [11] showed that a separable Banach space X can be embedded in an l p sum of finite-dimensional spaces for 1 < p < if and only if X is reflexive and every normalized weakly null tree has a branch which is equivalent to the usual l p basis. This result is closely related to the proof of the previous theorem.

11 L p MAXIMAL REGULARITY ON BANACH SPACES WITH A SCHAUDER BASIS On L r -regularity in L s spaces Let s [1, ). We consider our usual Cauchy problem: { u (t) + B(u(t)) = f(t) for t < T u() = where T (, + ), B is the infinitesimal generator of a bounded analytic semigroup on L s = L s ([, 1]) and f L ([, T ); L s ). Then we ask the following question: for what values of s and r in [1, ) does the solution u(t) = t e (t s)b f(s) ds necessarily satisfies u L ([, T ); L r )? Thus we introduce the following definition: Definition 4.1. Let r and s in [1, ). We say that (r, s) is a regularity pair if whenever B is the infinitesimal generator of a bounded analytic semigroup on L s = L s ([, 1]) and f L ([, T ); L s ), the solution u of { u (t) + B(u(t)) = f(t) for t < T u() = satisfies u L ([, T ); L r ). Notice that it follows from previous results ([3], [8] and [7]) that (s, s) is a regularity pair if and only if s =. This is extended by our next result: Theorem 4.. Let r and s in [1, ). Then (r, s) is a regularity pair if and only if r s =. Proof. It follows clearly from the work of De Simon [3], that if r s = then (r, s) is a regularity pair. So let now (r, s) be a regularity pair. Since L 1 does nat have (MRP) ([8]), we have that s > 1. Then, solving our Cauchy problem with B =, we obtain that r s. Thus we can limit ourselves to the case s > 1 and 1 r s. Then by the closed graph Theorem, for any B so that B is the infinitesimal generator of a bounded analytic semigroup on L s = L s ([, 1]), there is a constant C > such that for any f L ([, T ); L s ): u L (L s ) Cf L (L s ). Using the inclusion L s L r for r s, we can now state the following analogue of Theorem.1: Proposition 4.3. Let (E n, P n ) n 1 be a Schauder decomposition of L s. Assume that (r, s) is a regularity pair. Then there is a constant C > so that whenever

12 1 N. J. KALTON AND G. LANCIEN (u n ) N are such that u n [E n 1, E n ] then P n u n ε n L (L r ) C u n ε n L (L s ) Then our first step will be to show that the Haar system (h k ) satisfies some lower- estimates in L s in the following sense: Lemma 4.4. If there exists r s such that (r, s) is a regularity pair, then there is a constant C > such that for any normalized block basic sequence (v 1,..., v n ) of (h k ) and for any a 1,.., a n in C: a k v k L C a s k. Proof. We first observe that if 1 < p <, it follows from the work of J. Bretagnolle, D. Dacunha-Castelle and J.L. Krivine [1] on p-stable random variables that there is a sequence (e n ) n 1 in L 1 which is equivalent to the canonical basis of l p in any L q for 1 q < p. Thus (e n ) is weakly null in L s, and by a gliding hump argument, we may assume that (e n ) is actually a block basic sequence with respect with the Haar basis. If p =, then the Rademacher functions form a block basic sequence in every L q for 1 q <. Now assume the lemma is false. We pick a normalized block basic sequence (v 1,..., v n1 ) of (h k ) and a 1,.., a n1 in C so that n 1 a k v k L n 1 a s k = 1. Then pick m 1 N such that (v 1,.., v n1, e m1 ) is a block basic sequence of (h k ). By induction, we pick a normalized block basic sequence (v nj +1,..., v nj+1 ) of (h k ), a nj +1,.., a nj+1 in C and m j+1 N so that (v 1,.., v n1, e m1, v n1 +1,.., v nj+1, e mj+1 ) is a block basic sequence of (h k ) and n j+1 k=n j +1 a k v k L s 1 j n j+1 k=n j +1 a k = 1 j. So we can find (I k ) k 1 and (J k ) k 1 two sequences of finite intervals of N such that {I k, J k : k 1} is a partition of N and for all k 1, v k [h j, j I k ] and e mk [h j, j J k ]. Then set X k = [h j : j I k J k ]. Then (X k ) is an unconditional Schauder decomposition of L s. Each X k can be decomposed into X k = E k 1 E k, where E k 1 = [v k + e mk ], e mk E k and the corresponding projections are uniformly bounded. So, by Lemma 3., (E k ) k 1 is a Schauder decomposition of L s. We can now make use of of

13 L p MAXIMAL REGULARITY ON BANACH SPACES WITH A SCHAUDER BASIS 13 Proposition 4.3. If we decompose a k v k = a k (v k + e mk ) a k e mk in E k 1 E k, we obtain that there is a constant C > such that for all n 1: a k v k ε k L (L s ) C( a k ) 1/. Since (v k ) is an unconditional basic sequence in L s, there is a constant K > so that for all n 1: a k v k L K a s k, which is in contradiction with our construction. We now conclude the proof of Theorem 4.. The Haar basis of L s has a block basic sequence equivalent to the standard basis of l max(s,). Hence Lemma 4.4 shows that max(s, ) p whenever s < p < or p =. Thus s =. References [1] J. Bretagnolle, D. Dacunha-Castelle and J.L. Krivine, Lois stables et espaces L p, Annales de l Institut Henri Poincaré, (1966). [] T. Coulhon and D. Lamberton, Régularité L p pour les équations d évolution, Séminaire d Analyse Fonctionnelle Paris VI-VII ( ), [3] L. De Simon, Un applicazione della theoria degli integrali singolari allo studio delle equazioni differenziali lineare astratte del primo ordine, Rend. Sem. Mat., Univ. Padova (1964), 5-3. [4] G. Dore, L p regularity for abstract differential equations (In Functional Analysis and related topics, editor: H. Komatsu), Lect. Notes in Math. 154, Springer Verlag (1993). [5] T. Figiel, N. Tomczak-Jaegermann, Projections onto Hilbertian subspaces of Banach spaces, Israel J. Math. 33 (1979), no., [6] W.B. Johnson and M. Zippin, On subspaces of quotients of ( G n ) lp and ( G n ) c, Israel J. Math. 13 (197) [7] N.J. Kalton and G. Lancien, A solution to the problem of L p -maximal regularity, to appear in Math. Z. [8] C. Le Merdy, Counterexamples on L p -maximal regularity, Math. Z. 3 (1999), [9] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces, Vol.1, Sequence spaces, Springer, Berlin [1] V.D. Milman and G. Schechtman, Asymptotic theory of fninite-dimensional normed spaces, Springer Lecture Notes 1, Berlin [11] E. Odell and T. Schlumprecht, Trees and branches in Banach spaces, to appear.

14 14 N. J. KALTON AND G. LANCIEN [1] G. Pisier, Les inégalités de Kahane-Khintchin d après C. Borell, Séminaire sur la géometrie des éspaces de Banach, Ecole Polytechnique, Palaiseau, Exposé VII, [13] G. Pisier, Holomorphic semigroups and the geometry of Banach spaces, Ann. Math. 115 (198) [14] P.E. Sobolevskii, Coerciveness inequalities for abstract parabolic equations (translations), Soviet Math. Dokl. 5 (1964), [15] L. Weis, Operator valued Fourier Multiplier Theorems and Maximal L p Regularity, to appear. Department of Mathematics, University of Missouri-Columbia, Columbia, MO address: nigel@math.missouri.edu Equipe de Mathématiques - UMR 663, Université de Franche-Comté, F- 53 Besançon cedex address: glancien@math.univ-fcomte.fr