Interpolation of Bilinear Operators Between Quasi-Banach Spaces

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1 Positivity 10 (2006), c 2006 Birkhäuser Verlag Basel/Switzerland / , published online May 24, 2006 DOI /s x Positivity Interpolation of Bilinear Operators Between Quasi-Banach Spaces Loukas Grafakos 1 and Mieczys law Masty lo 2 Abstract. We study interpolation, generated by an abstract method of means, of bilinear operators between quasi-banach spaces. It is shown that under suitable conditions on the type of these spaces and the boundedness of the classical convolution operator between the corresponding quasi-banach sequence spaces, bilinear interpolation is possible. Applications to the classical real method spaces, Calderón-Lozanovsky spaces, and Lorentz-Zygmund spaces are presented. Mathematics Subject Classification (2000). 46B70; 46M35. Keywords. Interpolation, quasi-banach spaces, multilinear interpolation, bilinear operators, Calderón-Lozanovsky spaces, Lorentz-Zygmund spaces. 1. Introduction Motivated by applications in harmonic analysis, we are interested in interpolation of bilinear operators defined on products of quasi-banach spaces. The main aim of this paper is to prove interpolation theorems for bilinear operators on quasi- Banach spaces generated by certain interpolation methods. We study a problem for the abstract method of means as well as for the real interpolation method. Let us briefly outline the content of the paper. In Section 2 we establish notation and recall basic facts concerning quasi-banach spaces and interpolation. In Section 3 we introduce a notion of special type of convexity for bilinear operators between quasi-banach couples and we prove a bilinear interpolation theorem using the method of means, under the condition that the associated convolution operator is bounded on the parameter spaces involved in the construction of these methods. In view of a remarkable result of Kalton [13], the convexity parameters of the bilinear operators that take values in so called natural quasi-banach spaces, are nicely determined by the types of the domains of the quasi-banach spaces. We also 1 The author is supported by the National Science Foundation under grant DMS The author is supported by KBN Grant 1 P03A

2 410 L. Grafakos and M. Masty lo Positivity prove continuous inclusions between spaces generated by the method of means and the Calderón-Lozanovsky method applied to certain classes of couples of Banach lattices satisfying the upper or lower lattice estimates. We give applications in the context of weighted Orlicz spaces. In particular, we obtain that the method of means determined by the corresponding weighted quasi-banach spaces l p0 and l p1 and any couple of weighted quasi-banach lattices (L p0 (w 0 ),L p1 (w 1 )) coincide, up to equivalence of norms, with the Calderón-Lozanovsky space ϕ(l p0 (w 0 ),L p1 (w 1 )). In Section 4 we discuss applications of these results in the context of interpolation between quasi-banach spaces generated by the real method of interpolation as well as to Calderón-Lozanovsky spaces that include, in particular, Orlicz spaces. In Section 5 we discuss abstract K or J interpolation method of bilinear operators between quasi-banach spaces satisfying weaker convexity type conditions. These results are used for special weighted sequence spaces for which the classical convolution operator is bounded. As a consequence, bilinear interpolation theorems for Lorentz-Zygmund spaces are obtained. 2. Definitions and Notation A quasi-norm defined on a vector space X (over real or complex field K) isa map X R + such that (i) x > 0forx 0, (ii) αx = α x for α K, x X, (iii) x + y C( x + y ) for all x, y X, where C is a constant independent of x, y. Let 0 <p 1. We call a p-norm if we also have (i) x + y p x p + y p for all x, y X. A quasi-banach space X is said to be p-normable, 0<p 1, if there exists an equivalent p-norm on X and a constant C such that x x n C ( x 1 p + + x n p ) 1/p. for all x 1,..., x n X. An 1-normable space is simply called normable. While clearly any p-normable space is a quasi-normed space, a theorem of Aoki and Rolewicz (see [14]) asserts that any quasi-normed space X has an equivalent p-norm, where p satisfies C =2 1/p 1 with C is as in (iii), defined by x =inf ( x k p 1/p, X) k where the infimum is taken over all finite sequences {x k } X satisfying k x k =x. If is a quasi-norm (resp., p-norm) on X defining a complete metrizable topology, then X is called a quasi-banach space (resp., p-banach space). We shall use standard notation and notions from interpolation theory, as presented, e.g., in [2], [3]. Throughout this paper we will let (Ω,µ)=(Ω, Σ,µ)

3 Vol. 10 (2006) Interpolation of Bilinear Operators 411 be a complete σ-finite measure space and L 0 (µ) will denote, as usual, the space of equivalence classes of real valued measurable functions on Ω, equipped with the topology of convergence (in the measure µ) on sets of finite measure. By a quasi-banach lattice on Ω we mean a quasi-banach space X which is a subspace of L 0 (µ) such that there exists u X with u>0andif f g a.e., where g X and f L 0 (µ), then f X and f X g X. A quasi-banach lattice X is said to be maximal if its unit ball B X = {x; x 1} is a closed subset in L 0 (µ). In the special case when Ω = Z is the set of integers and µ is the counting measure then a quasi-banach lattice E on Ω is called a quasi-banach sequence space on Z, and in this case we denote by E aköthe dual space of E. If X is a quasi-banach lattice on Ω and w L 0 (µ) with w>0 a.e., we define the weighted quasi-banach lattice X(w) by x X(w) = xw X. Given 0 <p< and a quasi-banach lattice X let X p denote the p-convexification of X. Here X p consists of all x such that x p X and is equipped with the quasi-norm x = x p 1/p. Let 0 <t< and A =(A 0,A 1 ) be a couple of quasi-banach spaces. We equip A 0 + A 1 (resp., A 0 A 1 ) with the quasi-norm K(1,a)(resp.J(1,a)) where K(t, a) = K(t, a; A) andj(t, a) = J(t, a; A) are the functionals of J. Peetre, defined by and K(t, a; A) =inf{ a 0 A0 + t a 1 A1 ; a = a 0 + a 1 } J(t, a; A) = max{ a A0,t a A1 }. It is easy to see that if A j is p j -normable (j =0, 1) then both A 0 +A 1 and A 0 A 1 are p-normable with p = min{p 0,p 1 }. If X =(X 0,X 1 )andy =(Y 0,Y 1 ) are couples of quasi-banach spaces, we let L(X,Y ) be the quasi-banach space of all linear operators T : X Y (which means, as usual, that T : X 0 + X 1 Y 0 + Y 1 is linear and the restrictions T Xj are bounded operators from X j to Y j for j =0, 1). The space is equipped with the quasi-norm T X Y := max { T X0 Y 0, T X1 Y 1 }. We will deal with vector-valued quasi-banach sequence spaces. Let E be a quasi-banach sequence lattice on Z and let X be a quasi-banach space. The vector sequence x = {x n } n Z in X is called strongly E-summable if the corresponding scalar sequence { x n X } n Z is in E. We denote by E(X) thesetofall such sequences in X. This is a quasi-banach space under pointwise operations, and a natural quasi-norm on it is given by x E(X) := { x n X } E. It is easy to check that if E is p-banach and X is q-banach space then E(X) isr-banach with r = min{p, q}. Let X be a quasi-banach couple. A couple E =(E 0,E 1 ) of quasi-banach sequence lattices on Z is said to be a parameter of the method of means on X if E 0 E 1 l p for some 0 <p 1 such that the quasi-banach space X 0 + X 1 is p-normable. Throughout the paper, for such E =(E 0,E 1 )andx, the space

4 412 L. Grafakos and M. Masty lo Positivity denoted by J E (X) =J E0,E 1 (X) =X E0,E 1 is built by the method of means consisting of all x X 0 + X 1 which can be represented in the form x = u n (convergence in X 0 + X 1 ) n Z with {u n } E 0 (X 0 ) E 1 (X 1 ). We note that J E (X) is a quasi-banach space under the quasi-norm x := inf max { {u n } E0(X 0), {u n } E1(X 1)}, where the infimum is taken over all the above representations of x. In fact, the continuous inclusion E 0 E 1 l p implies that there exists a constant C>0such that u n p X 0+X 1 C {u n } E0(X 0) E 1(X 1) n Z for any {u n } E 0 (X 0 ) E 1 (X 1 ). Since X 0 +X 1 is p-normable, we conclude that the linear map J defined by J ({x n }):= n Z x n is continuous from E 0 (X 0 ) E 1 (X 1 ) into X 0 + X 1. Thus the quotient space ( E0 (X 0 ) E 1 (X 1 ) )/ ker(j ) is a quasi-banach space. Since it is isometrically isomorphic to J E (X), we conclude that J E (X) is also a quasi-banach space. 3. Main Results In this section we prove interpolation theorems for bilinear operators between spaces generated by the method of means. An operator T defined on a product of two quasi-banach spaces X Y and taking values in another quasi-banach space Z is called bilinear if it is linear in each of the two variables, and bounded, i.e., there is a constant C 0 such that for all x X and y Y we have T (x, y) Z C 0 x X y Y. The smallest C 0 so that the above inequality holds for all x X and y Y is called the norm of T and will be denoted by T X Y Z. Inspired by a remarkable result of Kalton [13] we introduce the following terminology: we say that a bilinear operator T : X Y Z between quasi-banach spaces is said to be s-bilinear convex (0 <s 1) if there exists a constant C>0 such that for all finite sequences {x j } n j=1 X and {y j} n j=1 Y, we have n T (x j,y j ) n Z C T X Y Z x j s X y j s Y j=1 In the case when s = 1, we say, in short, that T is bilinear convex. j=1 1/s.

5 Vol. 10 (2006) Interpolation of Bilinear Operators 413 The triple (X, Y, Z) of quasi-banach spaces is said to be s-bilinear (resp., bilinear) admissible whenever there exists C = C(X, Y, Z) > 0 such that any bilinear operator T : X Y Z is s-bilinear convex (resp., bilinear convex). In view of the result of Kalton ([13], p. 311), it follows that if X is a quasi- Banach space of type p, Y is a quasi-banach of type q, andz is a natural quasi- Banach space, then the triple (X, Y, Z) iss-bilinear admissible where 1/s = 1/p + 1/q. Let us recall that a quasi-banach space X is of type p (0 <p 2) if there exists a constant C>0sothat [ ( n 1/p ( n ) 1/p E ε k x k p)] C x k p, k=1 where {ε k } is any sequence of independent Bernoulli random variables with k=1 P (ε k =1)=P (ε k = 1) = 1/2. It is well-known that for 0 <p<1, X is of type p if (and only if) X is p-normable; if p>1andx is of type p, then X is a Banach space (see [14], p. 99 and p. 107). A quasi-banach space is called natural [15] if it is isomorphic to a subspace of an L-convex quasi-banach lattice. A quasi-banach lattice X is said to be L-convex if there exists 0 <ε<1sothatifu X, with u =1and0 x k u (1 k n) satisfy (x x n )/n (1 δ)u, then max 1 k n x k ε. The following lemma will be useful in the proof of the main result of this section. Lemma 3.1. Let X be a p-normable quasi-banach space and let {x k,m } be an infinite matrix in X with k, m Z. Assume that the series k Z x k,m k is unconditionally convergent for every m Z and m Z k Z x p k,m k <. Then the X double limit k M j N x k,j exists in X and lim M,N lim M,N k M j N x k,j = m Z ( ) x k,m k. Proof. Let u m := k Z x k,m k for m Z. Fixε>0. Since m Z u m p X < and the series k Z x k,m k converges unconditionally, there exists m 0 N such that and ( u m m m 0 m Z k Z m >m 0 k F m x k,m k k Z x k,m k ) p X <ε/2 p <ε/4, X

6 414 L. Grafakos and M. Masty lo Positivity where F m is any finite subset of Z. Furthermore, there exists k 0 N such that for any k m k 0 with m m 0,wehave X <ε/4. m m 0 k >k m x k,m k Let M and N be positive integers with M>k 0 and N>m 0 + k 0. We define two subsets A and B of Z Z by setting A = { (k, j); k M, j N } and B = { (k, j); k k 0, j + k m 0 }. For m Z we let F m = { k Z ; (k, m k) A} and k m = max { } k Z ; k F m. Since B A, we conclude that k m k 0, whenever m m 0. This implies that the two last inequalities hold for F m and k m just defined. It is easy to verify that x k,j = u m x k,m k k M j N m m 0 m m 0 k >k m + x k,m k. m >m 0 k F m Combining this identity with the above three norm inequalities, we deduce ( ) X x k,m k <ε. x k,j k M j N m Z k Z for M>k 0 and N>m 0 + k 0. This proves the assertion. Let X =(X 0,X 1 ), Y =(Y 0,Y 1 )andz =(Z 0,Z 1 ) be quasi-banach couples. We will say that T =(T 0,T 1 ) is a bilinear operator from X Y into Z, and write T B(X,Y ; Z) ift j : X j Y j Z j is a bounded bilinear operator (j =0, 1) and T 0 (x, y) =T 1 (x, y) for any x X 0 X 1 and y Y 0 Y 1. If additionally X, Y and Z are intermediate quasi-banach spaces with respect to X, Y and Z, respectively, then we say that T B(X,Y ; Z) extends to a bilinear operator from X Y into Z provided that T 0 has a bilinear extension from X Y into Z. Note that any (T 0,T 1 ) B(X,Y ; Z) defines a bilinear operator T 0 (resp., T 1 ) which will be called in the sequel a natural bilinear extension of (T 0,T 1 )from (X 0 + X 1 ) (Y 0 Y 1 )intoz 0 + Z 1 (resp., (X 0 X 1 ) (Y 0 + Y 1 ) Z 0 + Z 1 )by T 0 (x, y) :=T 0 (x 0,y)+T 1 (x 1,y) for any x = x 0 + x 1 and y Y 0 Y 1 (resp., T 1 (x, y) :=T 0 (x, y 0 )+T 1 (x, y 1 )for any x X 0 X 1 and y = y 0 + y 1 Y 0 + Y 1 with y 0 Y 0 and y 1 Y 1.Itiseasy to see that T 0 (resp., T 1 ) does not depend on the representations of x X 0 + X 1 (resp., y Y 0 + Y 1 ).

7 Vol. 10 (2006) Interpolation of Bilinear Operators 415 An operator T =(T 0,T 1 ) B(X,Y ; Z) is said to be (s 0,s 1 )-bilinear convex (0 <s 0,s 1 1) if there exists a constant C>0such that for any finite sequences {x j } X 0 X 1 and {y j } Y 0 Y 1 the following holds for k =0, 1 j T k (x j,y j ) Zk C T k Xk Y k Z k j x j s k Xk y j s k Yk 1/sk. In the case when s 0 = s 1 = s (resp., s 0 = s 1 = 1), we say, in short, that T is s-bilinear convex (resp., bilinear convex). Throughout the paper we will consider the convolution operator τ of sequences defined for x = {ξ k } and y = {η k } by τ(x, y) n = ξ k η n k, n Z. k Z We now state the main theorem of this section: Theorem 3.1. Let X =(X 0,X 1 ), Y =(Y 0,Y 1 ),andz =(Z 0,Z 1 ) be quasi-banach spaces and let T = (T 0,T 1 ) B(X,Y ; Z) be (s 0,s 1 )-bilinear convex. Assume that E j, F j and G j are quasi-banach sequence spaces on Z such that (E s0 0,Es1 1 ), (F s0 0,Fs1 1 ),and(gs0 0,Gs1 1 ) are parameters of the method of means on X, Y and Z, respectively. If the convolution operator τ is bounded from E j F j to G j (j =0, 1), then T extends to a bilinear operator T from X s E 0 0,Es 1 Y s 1 F 0 0,F s 1 1 with the norm estimate T max j=0,1 for some C j > 0(j =0, 1). [ Cj τ Ej F j G j T j Xj Y j Z j ] into Z s G 0 0,G s 1 1 Proof. Let x X := X s E 0 0,Es 1 and y Y := Y s 1 F 0 0,F s.forε>0pick{u 1 k} 1 E s0 0 (X 0) E s1 1 (X 1)and{v k } F s0 0 (Y 0) F s1 1 (Y 1) such that x = u k (convergence in X 0 + X 1 ), y = v k (convergence in Y 0 + Y 1 ) k Z k Z and {u k } s E j j (Xj) (1 + ε) x X, {v k } s F j j (Yj) (1 + ε) y Y. Since the convolution operator τ is bounded from E j F j G j and T =(T 0,T 1 ) is (s 0,s 1 )-bilinear convex, we immediately deduce that if S := T 0, then the series S(u k,v m k ) k Z converges unconditionally in both spaces Z 0 and Z 1 for all m Z, and thus also in Z 0 + Z 1. Furthermore if we set z m := k Z S(u k,v m k )form Z, we obtain for some C j > 0

8 416 L. Grafakos and M. Masty lo Positivity { z m Zj } s Gj j C j T j τ({ uk Xj Y j Z j sj X 0 }, { v k sj Y j }) 1/sj G j C j T j Xj Y j Z j τ Ej F j G j {u k } E s j (Xj) {v k } F s j (Yj) (1 + ε) 2 C j τ Ej F j G j T j Xj Y j Z j x X y Y. These calculations show that the sequence {z m } m Z lies in G s0 0 (Z 0) G s1 1 (Z 1)and {z m } s G 0 0 (Z 0) G s 1 1 (Z 1) (1+ε)2 max [C j T j Xj Y j Z j τ Ej F j G j ] x X y Y. j=0,1 Our hypothesis that (G s0 0,Gs1 1 ) is a parameter of the method of means on (Z 0,Z 1 ) implies that for some 0 <p 1 z m p Z 0+Z 1 <. m Z Applying Lemma 3.1, we deduce that the double limit T (x, y) := S(u k,v j ) lim m,n k m j n exists in Z 0 + Z 1 and T (x, y) = z m (convergence in Z 0 + Z 1 ). m Z Combining the above remarks yield T (x, y) Z := Z G s 0 0,G s 1 1 with T (x, y) Z C(1 + ε) 2 x X y Y, where C = max j=0,1 [C j T j Xj Y j Z j ] τ Ej F j G j. Since ε is arbitrary, to conclude it is enough to show that T defines a required bilinear extension of T. To see this recall that the natural extensions T 0 :(X 0 + X 1 ) (Y 0 Y 0 ) Z 0 + Z 1 and T 1 :(X 0 X 1 ) (Y 0 + Y 0 ) Z 0 + Z 1 are bilinear operators. This implies that the following limits exist in Z 0 + Z 1 for all k, j Z S(u k,v j )=T 0 (x, v j ), and lim m lim n k m S(u k,v j )=T 1 (u k,y). j n Combining this with the fact that double limit z := lim m,n k m S(u k,v j )existsinz 0 + Z 1, we easily obtain z = lim lim S(u k,v j ) = lim T 0 x, v j n m n j n k m j n j n

9 Vol. 10 (2006) Interpolation of Bilinear Operators 417 and z = lim m k m lim n S(u k,v j ) = lim j n This shows that the double limit T (x, y) := lim m,n k m j n m T 1 S(u k,v j ) u k,y. k m is independent of the representations of x = k Z u k and y = k Z v k. Therefore, we conclude that T is a bilinear operator from X Y into Z. Since T is an extension of T, the proof is complete. From the point of view of applications the following corollary is of independent interest. Corollary 3.1. Assume X =(X 0,X 1 ), Y =(Y 0,Y 1 ) are couples of Banach spaces of type 2 and Z =(Z 0,Z 1 ) is a couple of natural quasi-banach spaces. Let (E 0,E 1 ), (F 0,F 1 ) and (G 0,G 1 ) be parameters of the method of means on X, Y,andZ, respectively. If the convolution operator τ is bounded from E j F j to G j,for j =0, 1, then any T =(T 0,T 1 ) B(X,Y ; Z) extends to a bilinear operator T from X E0,E 1 Y F0,F 1 into Z G0,G 1 which satisfies the norm estimate T max j=0,1 for some C j = C(X j,y j,z j ) > 0(j =0, 1). [ Cj τ Ej F j G j T j Xj Y j Z j ] Proof. Using Kalton s [13] result, we conclude that both triples (X 0,Y 0,Z 0 )and (X 1,Y 1,Z 1 ) are bilinear admissible, thus Theorem 3.1 applies. We conclude this section by giving applications to methods of means generated by weighted quasi-banach sequence spaces determined by quasi-concave functions. Recall that a positive function ρ on (0, ) is said to be quasi-concave if ρ is non-decreasing and the function t ρ(t)/t is non-increasing. A quasi-concave function ρ is called a quasi-power if s ρ (t) =o(max{1,t}) ast 0andt, where s ρ (t) :=sup u>0 ( ρ(tu)/ρ(u) ) for t>0. Lemma 3.2. Let ρ be a quasi-power function and let Φ 0, Φ 1 be quasi-banach sequence spaces on Z such that Φ j l (j =0, 1). Then the following statements are true for E 0 =Φ 0 (1/ρ(q n )) and E 1 =Φ 1 (q n /ρ(q n )) and any q>1: (i) E 0 E 1 l r for any r>0. (ii) X E0,E 1 is a quasi-banach space for any quasi-banach couple X. (iii) If ρ(t) =t θ, 0 <θ<1 and Φ j = l pj, 0 <p j, then for any q>1 and any quasi-banach space X X E0,E 1 = X θ,p, where 1/p =(1 θ)/p 0 + θ/p 1.

10 418 L. Grafakos and M. Masty lo Positivity Proof. (i). Since ρ is a quasi-power function, it follows (see, e.g., [17], p ) ( ( { s} min 1, ρ(s)) r ds ) 1/r ρ(t) 0 t s for any r>0. In particular, this implies that for any q>1 ( ( C(r) := min{1,q n }ρ(q n ) ) ) 1/r r <. n Z Thus if Φ j l for j =0, 1, it follows that E 0 E 1 l (1/ρ(q n )) l (q n /ρ(q n )) = l (max{1/ρ(q n ),q n /ρ(q n )}). Consequently there exists a constant K>0such that for any {ξ n } E0 E 1 1, we have ξ n K min{ρ(q n ),ρ(q n )/q n } for all n Z. Thus {ξ n } lr C(r)K, i.e., E 0 E 1 l r. Clearly that (ii) follows by (i) and the remarks in Section 2. (iii). It is shown in [27] that if 1/p =(1 θ)/p 0 + θ/p 1, then the formula X E0,E 1 = X θ,p, holds with q = e. It is easy to see that the proof works for any q>1. Theorem 3.2. Assume that X =(X 0,X 1 ), Y =(Y 0,Y 1 ) and Z =(Z 0,Z 1 ) are quasi-banach spaces and T =(T 0,T 1 ) B(X,Y ; Z) is (s 0,s 1 )-bilinear convex. If p j,q j,r j [1, ) and 1/r j =1/p j +1/q j 1 for j =0, 1 and 1/p =(1 θ)/(s 0 p 0 )+ θ/(s 1 p 1 ), 1/q =(1 θ)/(s 0 q 0 )+θ/(s 1 q 1 ), 1/r =(1 θ)/(s 0 r 0 )+θ/(s 1 r 1 ) and 0 <θ<1, then T extends to a bilinear operator T from X θ,p Y θ,q into Z θ,r which satisfies the norm estimate T max C j T j Xj Y j Z j, j=0,1 for some constant C j > 0(j =0, 1). Proof. The hypothesis on the indices imply by Young s theorem that the convolution operator is bounded from l pj l qj into l rj, and thus also from l pj (a jθ ) l qj (a jθ )intol rj (a jθ ) for any a>0, j =0, 1. Applying Theorem 3.1 and Lemma 3.2, the required conclusion follows. We note that the triple (L p,l q,z)iss-bilinear admissible for any natural quasi-banach space Z, when 1/s =1/u +1/v, with (u, v) =(p, q) whenever 0 <p,q 2, (u, v) =(2, 2) whenever 2 p, q <, and (u, v) =(p, 2) whenever 0 <p 2 q<. Thus the obtained results may be applied to many bilinear operators such as bilinear multipliers. Recall that a bounded measurable function σ on R n R n gives rise to a bilinear operator W σ defined by W σ (f,g) = σ(ξ,η) f(ξ) ĝ(η)e 2πi x,ξ+η dξdη R n R n

11 Vol. 10 (2006) Interpolation of Bilinear Operators 419 where f,g are Schwartz functions and, denotes the inner product in R n. In this case σ is called the symbol of W σ. The study of such bilinear multiplier operators was initiated by Coifman and Meyer. A theorem of them [6] says that if 1 <p,q<, 1/r =1/p +1/q and the function σ on R n R n satisfies ξ α η β σ(ξ,η) C α,β ( ξ + η ) α β, for sufficiently large multi-indices α and β, then W σ extends to a bilinear operator from L p (R n ) L q (R n )intol r, (R n ) whenever r 1. Here as usual L r, denotes the space weak L r. This result was later extended to the range 1 >r 1/2 by Grafakos and Torres [8] and Kenig and Stein [16]. Multipliers that satisfy the Marcinkiewicz condition were studied by Grafakos and Kalton [9]. The first significant boundedness results concerning non-smooth symbols were proved by Lacey and Thiele [18], [19] who established that W σ, with σ(ξ,η) = sign(ξ+αη), α R\{0, 1} has a bounded extension from L p (R n ) L q (R n )tol r (R n )) when r>2/3. Extensions of this result were subsequently obtained by Gilbert and Nahmod [7]. Bilinear operators can also be defined on quasi-banach spaces, such as the Hardy spaces H p ; see for instance [10] for the action of bilinear Calderón-Zygmund operators on real Hardy spaces. We discuss here only a general application. Theorem 3.3. Let p j,q j,r j [1, ) and 1/r j =1/p j +1/q j 1 for j =0, 1 and let 1/p =(1 θ)/p 0 + θ/(2p 1 ), 1/q =(1 θ)/q 0 + θ/(2q 1 ), 1/r =(1 θ)/r 0 + θ/(2r 1 ) and 0 <θ<1. IfX =(X 0,X 1 ), Y =(Y 0,Y 1 ) are Banach spaces such that both X 0 and Y 0 are of type 2 and Z =(Z 0,Z 1 ) is a couple of natural quasi-banach spaces, then any T B(X,Y ; Z) extends to a bilinear operator T from X θ,p Y θ,q into Z θ,r which satisfies the norm estimate T C max T j Xj Y j Z j j=0,1 for some constant C>0. Proof. Using the aforementioned result of Kalton s, we conclude that the triple (X 0,Y 0,Z 0 ) is admissible and (X 1,Y 1,Z 1 )is1/2-admissible, thus Theorem 3.1 applies. Using the well-known results on interpolation by the real method between L p spaces (see, e.g., [2], Theorem 5.3.1) and the facts that any L p -space with 2 p< is of type 2 and any L q, with 0 <q< is L-convex, we obtain the following corollary for Lorentz spaces: Corollary 3.2. Let p j,q j,r j [1, ) and 1/r j =1/p j +1/q j 1 for j =0, 1 and also let 1/p =(1 θ)/p 0 + θ/p 1, 1/q =(1 θ)/q 0 + θ/q 1, 1/r =(1 θ)/r 0 + θ/r 1 and 0 <θ<1. If2 u j,v j < (j =0, 1) and 0 <t 0,t 1, then any bilinear operator T :(L u0,l u1 ) (L v0,l v1 ) (L t0,,l t1, ) has a bounded extension from L u,p L v,q into L t,r,where1/u =(1 θ)/u 0 + θ/u 1, 1/v =(1 θ)/v 0 + θ/v 1 and 1/t =(1 θ)/t 0 + θ/t 1.

12 420 L. Grafakos and M. Masty lo Positivity Corollary 3.2 yields, in particular, bounds for the bilinear Hilbert transforms and bilinear Calderón-Zygmund operators from products of Lorentz spaces into another Lorentz space. 4. Applications to Calderón-Lozanovsky Spaces In this section we prove a bilinear interpolation theorem for Calderón-Lozanovsky spaces. We show that under certain geometric conditions continuous inclusions hold between the method of means spaces and the Calderón-Lozanovsky spaces. In the case of quasi-banach couples of weighted L p -spaces, we obtain equalities of these spaces. Certain results in this direction for Banach spaces were shown in [24]. Following these ideas we extend some of these results for quasi-banach lattices (see, Theorem 4.1). Throughout this paper we denote by P (resp., U) the set of all functions ϕ : R + R + R + that are positive (resp., concave), non-decreasing in each variable, and homogeneous of degree one (that is, ϕ(λs, λt) = λϕ(s, t) for all λ, s, t 0). Let ϕ U and X =(X 0,X 1 ) be a couple of quasi-banach spaces on a measure space (Ω,µ). Following Calderón [4] and Lozanovsky [21], we define the space ϕ(x) =ϕ(x 0,X 1 ) of all x L 0 (µ) such that x = ϕ( x 0, x 1 ) for some x j X j, j =0, 1. We note that ϕ(x) is a quasi-banach (resp., Banach whenever X is a Banach couple) lattice equipped with the quasi-norm (resp., norm) x =inf { max{ x 0 X0, x 1 X1 }; x = ϕ( x 0, x 1 ) x j X j,j=0, 1 }. In particular, if we take ϕ(s, t) =s 1 θ t θ,0<θ<1, we obtain in this way the spaces X0 1 θ X1 θ introduced by Calderón [4]. The properties of the Banach lattice ϕ(x) have been studied in Lozanovsky (see [21] and references given therein). Following Kalton [13], a quasi-banach lattice X is said to be p-convex, 0 < p<, respectively q-concave, 0 <q<, if there exists a constant C>0such that ( ( n n ) 1/p x k p) 1/p C x k p respectively, k=1 ( n k=1 k=1 ) 1/q x k q C ( n k=1 x k q ) 1/q for every choice of elements x 1,..., x n X. A quasi-banach lattice X is said to be satisfy an upper p-estimate,0 < p <, respectively alowerq-estimate, 0 <q<, if there exists a constant C>0 such that for any choice of elements x 1,...,x n X.

13 Vol. 10 (2006) Interpolation of Bilinear Operators 421 respectively sup x k ( n ) 1/p C x k p, 1 k n k=1 ( n ) 1/q x k q C n x k. k=1 It is clear that if X is maximal, then the notion of an upper p-estimation is equivalent to the condition that sup x k ( ) 1/p C x k p k 1 k=1 holds for all disjointly supported infinite sequences. We note that p-convexity implies p-normability and this in turn yields an upper p-estimate. For p = 1, 1-convexity is equivalent to normability (as a Banach lattice). In the sequel, a function ϕ Pis said to be a quasi-power provided that the function t ϕ(1,t) is a quasi-power. It follows by Lemma 3.2 that if ϕ is a quasi-power, then for any 0 <p 0,p 1 the couple (E 0,E 1 )=(l p0 (1/ϕ(1, 2 n )), l p1 (2 n /ϕ(1, 2 n ))) is a parameter of the method of means on any couple (X 0,X 1 ) of quasi-banach spaces. The space (X 0,X 1 ) E0,E 1 is denoted by ϕ(x 0,X 1 ) p0,p 1. If X is an intermediate quasi-banach space with respect to a quasi-banach couple X =(X 0,X 1 ), we define its Gagliardo completion X c to be the space of all limits in X 0 + X 1 of sequences {x n } that are bounded in X, equipped with the quasi-norm x X c = inf sup x n X, {x n} n 1 where {x n } X has the same meaning as above. It is easy to check that if X is a maximal quasi-banach lattice on a measure space, then its Gagliardo completion X c equals to X. We have the following result: Theorem 4.1. Assume that (X 0,X 1 ) is a couple of quasi-banach lattices on a measure space (Ω,µ). Then the following continuous inclusions hold for any quasipower function ϕ U: (i) If X j satisfy an upper p j -estimate (j =0, 1), then k=1 ϕ(x 0,X 1 ) p0,p 1 ϕ(x 0,X 1 ) c. (ii) If X j is maximal and satisfy an upper p j -estimate (j =0, 1), then ϕ(x 0,X 1 ) p0,p 1 ϕ(x 0,X 1 ).

14 422 L. Grafakos and M. Masty lo Positivity (iii) If X j satisfy a lower q j -estimate (j =0, 1), then ϕ(x 0,X 1 ) ϕ(x 0,X 1 ) q0,q 1. Proof. (i) Let x ϕ(x 0,X 1 ) p0,p 1 with x < 1. Then m x = u k (convergence in X 0 + X 1 ) lim m,n k= n with {u n /ϕ(1, 2 n )} lp0 (X 0) 1and {2 n u n /ϕ(1, 2 n )} lp1 (X 1) 1. Since ϕ is a quasi-power, the series n Z u n is r-absolutely convergent in X 0 + X 1 for some 0 <r 1. Thus, in particular, { u n (ω) } l 1 for almost all ω Ω. Since X j satisfy an upper p j -estimate (j =0, 1), there are positive constants C 0 and C 1 such that for and x 0 n := sup k n x 1 n := sup k n u k ϕ(1, 2 k ) X 0 2 k u k ϕ(1, 2 k ) X 1 and we have x 0 n X0 C 0 and x 1 n X1 C 1 for all n N. We apply Carlson s inequality (see [12], Corollary 3.1) which states that for any quasi-power function ϕ U there exists a constant C>0such that for any finite positive sequence {a n } the following inequality holds ( a k Cϕ sup k k a k ϕ(1, 2 k ), sup k 2 k a k ϕ(1, 2 k ) Combining this inequality with the above estimates yields n u k Cϕ(x 0 n,x 1 n), k= n i.e., n k= n u k ϕ(x 0,X 1 ) with n k= n u k C maxj=0,1 C j for all n N. Since lim n u k x n X0+X =0 1 k= n the proof of (i) is complete. (ii) The result follows by a minor modification of the proof of (i). (iii) Let 0 x ϕ(x) and x ϕ(x) < 1. Then x = ϕ(x 0,x 1 ) for some 0 x j X j such that x j Xj < 1, j =0, 1. Since ϕ is a quasi-power function, it follows that the support of x is contained in the intersection of the supports of x 0 and x 1. Hence, without loss of generality, we may suppose that x, x 0,x 1 are not equal to zero on the domain Ω. ).

15 Vol. 10 (2006) Interpolation of Bilinear Operators 423 Define for any k Z, A k = { ω Ω; 2 k x 1 (ω)/x 0 (ω) < 2 k+1} and y k = xχ Ak,u k = x 0 χ Ak, v k = x 1 χ Ak. Clearly y k X 0 X 1. Is is easily seen that for any k Z, wehave y k 2ϕ(1, 2 k ) u k and y k ϕ(1, 2k ) 2 k v k. This implies that for any positive integer n the following estimates hold 0 y k ϕ(1, 2 n ) y k ϕ(1, 2 k ) 2ϕ(1, 2 n ) x 0, k n k n 0 y k ϕ(1, 2n ) 2 n k n k n Combining these estimates, we obtain N x y k C X0+X 1 k= M 2 k y k ϕ(1, 2 k ) ϕ(2 n, 1) x 1. ( M 1 k= X0 y k + k=n+1 y k X1 ) 2Cϕ(1, 2 M 1 )+2Cϕ(2 N 1, 1) for any positive integers M and N. Since ϕ is quasi-power, the right hand of the above inequality approaches 0 whenever M,N This implies that the series k Z y n converges to x in X 0 + X 1. Furthermore, by the fact that {A n } is a sequence of pairwise disjoint measurable subsets whose union is equal to Ω, we have y k ϕ(1, 2 k ) 2u k 2x 0 k k and 2 k y k ϕ(1, 2 k ) v k x 1. k k Now assume that X j satisfy a lower q j -estimate (j =0, 1). Combining the above inequalities yields { xk /ϕ(1, 2 k ) } l q0 (X 0 )and { 2 k x k /ϕ(1, 2 k ) } l q1 (X 1 ). Consequently x ϕ(x 0,X 1 ) q0,q 1. Since any L p -space is maximal and satisfies both a lower p-estimate and an upper p-estimate for any 0 <p<, the following corollary is an immediate consequence of Theorem 4.1.

16 424 L. Grafakos and M. Masty lo Positivity Corollary 4.1. If ϕ U is a quasi-power function, then for any 0 <p 0,p 1 < and weights w 0 and w 1 ϕ(l p0 (w 0 ),L p1 (w 1 )) p0,p 1 = ϕ(l p0 (w 0 ),L p1 (w 1 )). It is well known (see, e.g., [26]) that for any ϕ U and any couple (L p0 (w 0 ), L p1 (w 1 )) on (Ω,µ) with 0 < p 0 < p 1, the Calderón-Lozanovsky space ϕ(l p0 (w 0 ),L p1 (w 1 )) coincides up to equivalence of norms with the generalized Orlicz space of all f L 0 (µ) such that M (( w 1/p1 1 w 1/p0 0 ) q f /λ ) (w 0 /w 1 ) q dµ Ω for some λ>0. Here 1/q =1/p 0 1/p 1 and M is an Orlicz function such that M 1 (t) ϕ(t 1/p0,t 1/p1 )fort>0. We conclude this section by showing a particular application of Theorems 4.1 and 3.1 to bilinear operators on Orlicz spaces. For others results we refer to [23]. Theorem 4.2. Let ϕ 0,ϕ 1,ϕ U be quasi-power functions such that ϕ(1,st) Cϕ 0 (1,s) ϕ 1 (1,t) for some C > 0 and all s, t > 0. If 1 p j,q j <, 1 r j (j = 0, 1) are such that 1/r j = 1/p j +1/q j 1, then any operator T : (L p0 (u 0 ),L p1 (u 1 )) (L q0 (v 0 ),L q1 (v 1 )) (L r0 (w 0 ),L r1 (w 1 )) extends to a bounded bilinear operator from ϕ 0 (L p0 (u 0 ),L p1 (u 1 )) ϕ 1 (L q0 (v 0 ),L q1 (v 1 )) into ϕ(l r0 (w 1 ),L r1 (w 1 )). Proof. It is easy to see that if the convolution operator τ is bounded from E F into G, then it is bounded from E(u) F (v) intog(w) whenever there exists a constant C>0such that the sequences u = {u k }, v = {v k } and w = {w k } satisfy the condition w n Cu n k v k for some C>0and all k, n Z. Our hypothesis implies that the convolution operator is bounded from l pj l qj into l rj (j =0, 1), and thus Theorem 3.1 and Corollary 4.1 apply. 5. Bilinear Interpolation Between J and K-Method Spaces In this section we show that bilinear interpolation is possible under weaker assumptions on quasi-banach couples. We recall that if X =(X 0,X 1 ) is a couple of quasi- Banach spaces and E is a parameter of the K-method (i.e., E is a quasi-banach sequence space on Z such that {min(1, 2 n )} E), then the K-method space is a quasi-banach space K E (X) (denoted also by X E ) consists of all x X 0 + X 1 such that {K(2 n,x; X)} E. The space is equipped with the quasi-norm x := {K(2 n,x; X)} E. In what follows the method of means J E (X) (on a quasi-banach couple X) generated by a couple E =(E,E(2 n )) is called J-method space (on X) andis denoted by J E (X) ande is called a parameter of the J-method on X (resp., a parameter of J-method if it is a parameter of J-method on any quasi-banach couple X). For the study of abstract J and K Banach method spaces we refer to [5] and [3].

17 Vol. 10 (2006) Interpolation of Bilinear Operators 425 In the spirit of the previous terminology we introduce the following: Let X =(X 0,X 1 ), Y =(Y 0,Y 1 ), and (Z 0,Z 1 ) be couples of quasi-banach spaces. We say that a bilinear operator T =(T 0,T 1 ):X Y Z is right (resp., left) s-convex (0 <s 1) if there exists a constant C>0 such that, for any x X 0 X 1 and any finite sequence {y j } Y 0 Y 1 (resp., any y Y 0 Y 1 and any finite sequence {x j } n j=1 X 0 X 1 )wehave ( resp., n T 0 (x, y j ) ( n Z0 C T 0 X0 Y 0 Z 0 x X0 y j s ) 1/s Y 0 j=1 j=1 n j=1 T 1 (x j,y) ( n Z1 C T 1 X1 Y 1 Z 1 y Y1 x j s ) 1/s ) X 1. In the case when s = 1, we simply say that T is right (resp., left) convex. Clearly if T B(X,Y ; Z) is(s, s)-convex, then it is right and left s-convex. In particular if X =(X 0,X 1 ), Y =(Y 0,Y 1 ) are couples of Banach spaces of type 2 and Z is any couple of natural quasi-banach spaces, then any operator T B(X,Y ; Z) isconvex. If a quasi-banach space X is intermediate with respect to a quasi-banach couple (X 0,X 1 ), the closed convex hull of X 0 X 1 in X is denoted by X. The following theorem is an extension of the classical Lions-Peetre result (see, e.g., Bergh and Löfstrom[2], Theorem and Exercise , or Lions and Peetre [20], Zafran [28] and Astashkin [1]). Theorem 5.1. Let X =(X 0,X 1 ), Y =(Y 0,Y 1 ) and Z =(Z 0,Z 1 ) be quasi-banach space. Assume that E s is a parameter of the J-method space on X and both F s, G s with 0 <s<1 are parameters of the K-method such that the convolution operator τ is bounded from E F into G. Then the following statements are valid: (i) If T B(X,Y ; Z) is left s-convex, then it extends to a bilinear operator from J E s(x) K F s(y ) into K E s(z). (ii) If T B(X,Y ; Z) is right s-convex, then it extends to a bilinear operator from J E s(x) K F s(y ) into K G s(z). Proof. (i) Fix ε>0, x J E s(x) andy Y 0 Y 1. Then there exists {u k } X 0 X 1 such that {J(2 k,u k ; X)} E with j=1 x JE(X) (1 + ε) {J(2k,u k ; X)} E. Let T 0 :(X 0 +X 1 ) (Y 0 Y 1 ) Z 0 +Z 1 be a natural bilinear extension of T. Since T is s-convex, there exists a constant C>0 such that for any n Z, y 0 Y 0

18 426 L. Grafakos and M. Masty lo Positivity and y 1 Y 1 with y 0 + y 1 = y, we have with T := max j=0,1 T j Xj Y j Z j K(2 n,t 0 (x, y); Z) s T 0 (x, y 0 ) s Z 0 +2 sn T 0 (x, y 1 ) s ( Z 1 ) (C T ) s u k s X 0 y 0 s Y 0 +2 sn u k s X 1 y 1 s Y 1 k Z 2 1 s (C T ) s( J(2 k,u k ; X) s ( y 0 Y0 +2 n k y k Y1 ) s). k Z Taking the infimum over all decompositions y = y 0 + y 1, we obtain K(2 n,t 0 (x, y); Z) s 2 1 s (C T ) s J(2 k,u k ; X) s K(2 n k,y; Y ) s. k Z Combining these relations with the fact that the convolution operator τ is bounded from E F into G and ε is arbitrary we obtain T 0 (x, y) KG s (Z) 2 1 s 1 C T τ E F G x JE s (X) y K F s (X) This concludes the proof of (i). Using a natural bilinear extension T 1 :(X 0 X 1 ) (Y 0 + Y 1 ) Z 0 + Z 1, we prove (ii) in a similar way. From the point of view of applications in the above theorem the case E = F = G seems interesting. Let us remark that from the proof of Lemma 3.2, it follows that for any quasi-power function ρ and 0 < p the weighted quasi-banach sequence space E = l p (1/ρ(2 n )) is a parameter of both the J and K-methods of interpolation. Moreover, we have J E (X) =K E (X) for any quasi-banach couple X =(X 0,X 1 ) (see [11], [22]), and therefore we write X ρ,p instead of J E (X) or K E (X). It is easy to check that X 0 X 1 is dense in X ρ,p whenever 0 <p<. Combining these remarks with Theorem 5.1, we obtain immediately the following: Corollary 5.1. If 1 p, q, r satisfy 1/r =1/p +1/q 1, then any left or right convex operator T B(X,Y ; Z) extends to a bilinear operator from X θ,p Y θ,q into Z θ,r for any 0 <θ<1. Corollary 5.2. If 0 < p and ρ is a quasi-power function such that 1 C(ρ) :=sup n Z ρ(2 n { ρ(2 k ) ρ(2 n k ) } ) k <, (lp) then any left or right convex operator T B(X,Y ; Z) extends to a bilinear operator from X ρ,p Y ρ,p into Z ρ,p. Proof. The proof closely follows the proof of the result of [1] for the Banach case. Let E := l p (1/ρ(2 n )), 0 <p.fixx = {ξ n } E and y = {η n } E. Then we have τ(x, y)n ξ k η n k ρ(2 k ) ρ(2 n k ρ(2 k ) ρ(2 n k )) ) k Z { ξk η n k { ρ(2 k ) ρ(2 }k n k ρ(2 k ) ρ(2 n k ) } lp k. (lp)

19 Vol. 10 (2006) Interpolation of Bilinear Operators 427 This implies that τ(x, y) E C(ρ) x E y E. Therefore the convolution operator τ is bounded from E E into E, and Theorem 5.1 applies. We conclude the paper by discussing some applications to Lorentz-Zygmund spaces. Let (Ω,µ) be a measure space. Let 0 <p,0<q,andγ R. Recall that the Lorentz-Zygmund space L p,q (log L) γ is defined as the space of all functions that satisfy ( µ(ω) ( f p,q,γ := t 1/p (1 + log t ) γ f (t) ) q dt ) 1/q < 0 t for 0 <q< and ( f p,,γ := sup t 1/p (1 + log t ) γ f (t) ) < 0<t<µ(Ω) whenever q =. This space coincides with the classical Lorentz space L p,q if γ =0. In the next and final result all considered couples are defined on any finite measure space. Theorem 5.2. Assume that 2 p 0 <p 1 <, 2 q 0 <q 1 <, 0 <r 0 <r 1 and 1/p =(1 θ)/p 0 + θ/p 1, 1/q =(1 θ)/q 0 + θ/q 1, 1/r =(1 θ)/r 0 + θ/r 1 with 0 <θ<1. IfT :(L p0,l p1 ) (L q0,l q1 ) (L r0,l r1 ), then T has a bounded extension from L p,s (log L) γ L q,s (log L) γ to L r,s (log L) γ for any γ < 1 and 0 <s. Proof. It is easy to check that if f(t) = t α (1 + log t ) γ, where 0 < α <, γ R, then s f (t) =t α (1 + log t ) γ.thusf is a quasi-power function whenever 0 <α<1, γ R. Further it is well known (see, e.g., [25]) that if 0 <v,u 0,u 1, u 0 u 1 and f(t) =t θ (1 + log t ) γ (0 <θ<1, γ R), then (L u0,l u1 ) f,v = L u,v (log L) γ where 1/u =(1 θ)/u 0 + θ/u 1. Now, following [1] we define a function ψ by ψ(t) =t a ln c (C 1 /t) for0<t 1 and ψ(t) =t b ln d (C 2 t)fort>1, where 0 <a<b<1, c>1, d>1andc 1 >e c/a, C 2 >e d d/(1 b). Then ψ is a quasi-power function and for 0 <p the function ρ defined by ρ(t) = t/ψ(t) satisfies C(ρ) =sup n Z 1 ρ(2 n ) { ρ(2 k ) ρ(2 n k ) } k <. (lp) Observe that if A =(A 0,A 1 ) is a quasi-banach space such that A 1 A 0, then for any quasi-power function ρ and 0 < p the real method space (A 0,A 1 ) ρ,p consists of all a A 0 equipped with the quasi-norm ( 1 ( K(t, a; A) ) p dt ) 1/p. a = ρ(t) t 0

20 428 L. Grafakos and M. Masty lo Positivity Now, fix 0 <θ<1andγ< 1. Taking a =1 θ and c = γ, we conclude that the real method space (A 0,A 1 ) ρ,p generated by a quasi-power function ρ = t/ψ(t) defined above depends only on ρ restricted to (0, 1). Clearly on the interval (0, 1) the function ρ is equivalent to f(t) = t θ (1 + log t ) γ, thus using the interpolation formula of Merucci [25] and the fact that our hypothesis 2 p 0 <, 2 q 0 < implies by Kalton s result [13] (by L pj and L qj, j =0, 1areoftype 2) that T is bilinear convex, we may apply Corollary 5.2 to conclude the proof of the theorem. References [1] S.V. Astashkin, On Interpolation of Bilinear Operators by the Real Method of Interpolation, Mat. Zametki, 52 (1992), (Russian). [2] J.Bergh,J.Löfström, Interpolation Spaces. An Introduction, Springer, Berlin (1976). [3] A.Yu. Brudnyi, N.Ya. Krugljak, Interpolation Functors and Interpolation Spaces I, North-Holland, Amsterdam, (1991). [4] A.P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math., 24 (1964), [5] M. Cwikel, J. Peetre, Abstract K and J spaces, J. Math. Pures. Appl., 60 (1981), [6] R.R. Coifman, Y. Meyer, Commutateurs d intègrales singulières et opérateurs multilinéaires, Ann. Inst. Fourier (Grenoble), 28 (1978), [7] J. Gilbert, A. Nahmod, Bilinear operators with non-smooth symbols, J. Fourier Anal. Appl. 7 (2001), [8] L. Grafakos, R. Torres, Multilinear Calderón Zygmund theory, Adv. in Math. 40 (1996), [9] L. Grafakos, N.J. Kalton, The Marcinkiewicz multiplier condition for bilinear operators, Studia Math. 146 (2001), [10] L. Grafakos, N.J. Kalton, Multilinear Calderón-Zygmund operators on Hardy spaces, Collect. Math., 52 (2001), [11] J. Gustavsson, A function parameter in connetion with interpolation of Banach spaces, Math. Scand., 42 (1978), [12] J. Gustavsson, J. Peetre, Interpolation of Orlicz spaces, Studia Math., 60 (1977), [13] N.J. Kalton, Convexity conditions for non-locally convex lattices, Glasgow Math. J., 25 (1984), [14] N.J. Kalton, N.T. Peck, J.W. Roberts, An F -space Sampler, London Math. Soc. Lecture Notes 89, Cambridge University Press, (1985). [15] N.J. Kalton, Plurisubharmonic functions on quasi-banach spaces, Studia Math. 84 (1986), [16] C. Kenig, E.M. Stein, Multilinear estimates and fractional integration, Math. Res. Lett. 6 (1999), [17] S.G. Krein, Yu.I. Petunin, E.M. Semenov, Interpolation of Linear Operators, Nauka, Moscow, 1978 (in Russian); English transl.: Amer. Math. Soc., Providence, (1982).

21 Vol. 10 (2006) Interpolation of Bilinear Operators 429 [18] M.T. Lacey, C.M. Thiele, L p bounds for the bilinear Hilbert transform, p>2, Ann. of Math., 146 (1997), [19] M.T. Lacey, C.M. Thiele, On Calderón s conjecture, Ann. of Math., 149 (1999), [20] J.L. Lions, J. Peetre, Sur une classe d espaces d interpolation, Inst. Hautes Etudes Sci. Publ. Math., 19 (1964), [21] G.Ya. Lozanovsky, On some Banach lattices IV, Sibirsk. Mat. Z. 14 (1973), (in Russian); English transl.: Siberian. Math. J. 14 (1973), [22] M. Masty lo, On interpolation of some quasi-banach spaces, J. Math. Anal. Appl. 147 (1990), [23] M. Masty lo, On interpolation of bilinear operators, J. Funct. Anal., 214 (2004), [24] M. Masty lo, Interpolation methods of means and orbits, Studia Math., 17 (2005), [25] C. Merucci, Application of interpolation with function parameter to Lorentz, Sobolev and Besov spaces, In: Interpolation Spaces and Allied Topics in Analysis, Lecture Notes in Math (1984), [26] P. Nilsson, Interpolation of Banach lattices, Studia Math., 82 (1985), [27] Y. Sagher, Interpolation of r-banach spaces, Studia Math., 26 (1966), [28] M. Zafran, A multilinear interpolation theorem, Studia Math., 62 (1978), Loukas Grafakos Department of Mathematics University of Missouri Columbia, MO USA loukas@math.missouri.edu Mieczys law Masty lo Faculty of Mathematics and Computer Science A. Mickiewicz University and Institute of Mathematics Polish Academy of Science (Poznań) Umultowska Poznań Poland mastylo@amu.edu.pl Received 16 January 2004; accepted 21 August 2005 To access this journal online: