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Interpolation of Bilinear Operators and Compactness

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Interpolation of Bilinear Operators and Compactness INTERPOLATION OF BILINEAR OPERATORS AND COMPACTNESS E. BRANDANI DA SILVA1 and D. L. FERNANDEZ2 1Universidade Estadual de Maring´a- UEM, Departamento de Matem´atica - Av. Colombo 5790, Maring´a- PR, Brazil - 87020-900 email: [email protected] 2Universidade Estadual de Campinas - Unicamp, Instituto de Matem´atica - Caixa Postal 6065, Campinas - SP, Brazil - 13083-859 email: [email protected] Abstract. The behavior of bilinear operators acting on interpolation of Banach spaces for the ρ method in relation to the compactness is analyzed. Similar results of Lions-Peetre, Hayakawa and Person’s compactness theorems are obtained for the bilinear case and the ρ method. ——————————– Key words and phrases: interpolation, Banach spaces, bilinear operators, compactness 2000 Mathematics subject classification: 46B70, 46B50, 46M35 1 Introduction The study of compactness of multilinear operators for interpolation spaces goes back to A.P.Calder´on [4, pp.119-120]. Under an approximation hy- pothesis, Calder´on established an one-side type general result, but re- stricted to complex interpolation spaces. arXiv:1206.0017v1 [math.FA] 31 May 2012 For the real method, if E = (E0, E1), F = (F0, F1) and G = (G0, G1) are Banach couples, a classical result by Lions-Peetre assures that if T is a bounded bilinear operator from (E0 + E1) × (F0 + F1) into G0 + G1, whose restrictions T |Ek × Fk (k = 0, 1) are also bounded from Ek × Fk into 1 Gk (k = 0, 1) , then T is bounded from Eθ,p;J × Fθ,q;J into Gθ,r;J , where 0 <θ< 1 and 1/r = 1/p + 1/q − 1 . Lately several authors have obtained new and more general results for interpolation of bilinear and multilinear operators, for example see [9], [12] and [13]. On the other hand, the behaviour of compact multilinear operators under real interpolation functors or more general functors does not seem to have been yet investigated. This is our main subject in this work. After some preliminaries on interpolation of linear and bilinear operators, generalizations of Lions-Peetre compactness theorems [11, Theorem V.2.1] (the one with the same departure spaces) and [11, Theorem V.2.2] (the one with the same arriving spaces) will be stated. The proof of the first one is an adaptation of the original proof, but the later requires an involved argument. Thereafter, a two-side result for general interpolation functors of type ρ, with the additional cost of an approximation hypothesis on the departure Banach couples, will be then given. Thus, a theorem of Hayakawa type (i.e. a two-side result without approximation hypothesis) will be obtained. The point at this issue is that the Hayakawa type theorem is nothing but a corollary of the result with approximation hypothesis. Consequently, an one-side result holds for ordered Banach couples. Finally, as a consequence of the second Lemma of Lions-Peetre type, a compactness theorem of Persson type is obtained. To avoid ponderous notations we have restricted ourselves to the bilinear case. A generalization for the ρ method of the Lions-Peetre’s bilinear theorem will be also provided in this work. This work was published at ”Nonlinear Analysis: Theory, Methods and Applications, Volume 73, Issue 2, 2010, Pages 526-537”. Since there are some gaps in the original proof of Theorem 4.3, we give a new proof. For this, we change the Lemma 4.2. 2 Preliminaries on Interpolation 2.1 Interpolation functors. A pair of Banach spaces E = (E0, E1) is said to be a Banach couple if E0 and E1 are continuously embedded in some Hausdorff linear topological space E. Then we can form their intersection E0 ∩ E1 and sum E0 + E1; it can be seen that E0 ∩ E1 and E0 + E1 become 2 Banach spaces when endowed with the norms || x ||E0∩E1 = max{ ||x||E0 , ||x||E1 }, and || x ||E0+E1 = inf {||x0||E0 + ||x1||E1 }, x=x0+x1 respectively. We shall say that a Banach space is an intermediate space in respect to a Banach couple E = (E0, E1) if .E0 ∩ E1 ֒→ E ֒→ E0 + E1 .(The hookarrow ֒→ denotes bounded embeddings) Let E = (E0, E1) and F = (F0, F1) be Banach couples. We shall denote by L(E, F) the set of all linear mappings from E0 + E1 into F0 + F1 such that T |EK is bounded from Ek into Fk, k = 0, 1. By an interpolation functor F we shall mean a functor which to each Banach couple E = (E0, E1) associates an intermediate space F(E0, E1) between E0 and E1, and such that T |F(E0,E1) ∈ L(F(E0, E1), F(F0, F1)), for all T ∈ L(E, F). The interpolation functors which we shall consider depend on function parameters. 2.2 The function parameters. By a function parameter ρ we shall mean a continuous and positive function on R+. We shall say that a function parameter ρ belongs to the class B, if it satisfies the following conditions: (1) ρ(1) = 1, and ρ(st) (2) ρ(s) = sup < +∞, s> 0. t>0 ρ(t) Also, we shall say that a function parameter ρ ∈ B belongs to the class B+− if it satisfies ∞ 1 dt (3) min(1, )ρ(t) < +∞. t t Z0 3 From (1)-(3) we see that B+− is contained in Peetre‘s class P+−, i.e. the class of pseudo-concave function parameters which satisfies (4) ρ(t)= o(max(1,t)). (See Gustavsson [6] and Gustavsson-Peetre [7].) θ The function parameter ρθ(t) = t , 0 ≤ θ ≤ 1, belongs to B. It corre- +− sponds to the usual parameter θ. Further, ρθ ∈ B if 0 <θ< 1, but ρ0, +− ρ1 6∈ B . To control function parameters we shall need to recall the Boyd indices (see Boyd [2], [3] and Maligranda [12]). 2.3 The Boyd indices. Given a function parameter ρ ∈B, the Boyd in- dices αρ and βρ of the submultiplicative function ρ are defined, respectively, by log ρ(t) (5) αρ = sup , 1<t<∞ log t and log ρ(t) (6) βρ = sup . 0<t<1 log t The indices αρ and βρ are real numbers with the followings properties ∞ dt (7) α < 0 ⇐⇒ ρ(t) < +∞, ρ t Z1 and 1 dt (8) β > 0 ⇐⇒ ρ(t) < +∞. ρ t Z0 For the above mentioned function parameters ρ0 and ρ1 we have αρ0 < 0 θ and βρ1 > 0, respectively. For the function parameter ρθ(t)= t , 0 <θ< 1, we have αρθ = βρθ = θ. It can be proved that for all ρ ∈B, with βρ > 0 (αρ < 0, respectively) there exists an increasing (decreasing, respectively) function parameter ρ+ +− (ρ−, respectively) equivalent to ρ. Hence, if ρ ∈B it can be considered an increasing parameter, and ρ(t)/t a decreasing parameter. Furthermore, ρ can be considered non-decreasing, and ρ(t)/t non-increasing. Consequently, if ρ ∈B+− and 0 <q ≤∞, we have 4 −1 q ||ρ (t) min(1,t) ||L∗ < ∞. 2.4 Interpolation with function parameters. Let {E0, E1} and {F0, F1} be Banach couples and let L({E0, E1}, {F0, F1}) be the family of all linear maps T : E0 + E1 → F0 + F1 such that T |Ek is bounded from Ek to Fk, k = 0, 1. If E and F are intermediate spaces with respect to {E0, E1} and {F0, F1}, respectively, we say that E and F are interpolation spaces of +− type ρ where ρ ∈ P if given any T ∈ L({E0, E1}, {F0, F1}) we have ||T || ||T || ≤ C||T || ρ 1 , L(E,F ) 0 ||T || 0 for all T ∈ L({E0, E1)}, {F0, F1}), where ||T ||k = ||T ||L(Ek,Fk), (k = 0, 1) and C > 0 is a constant. Let {E0, E1} be a Banach couple. The J and K functionals are defined by J(t,x)= J(t,x; E) = max{||x||E0 , t ||x||E1 }, x ∈ E0 ∩ E1, K(t,x)= K(t,x; E)= inf {||x0||E0 + t ||x1||E1 }, x=x0+x1 respectively, where in K(t,x), x0 ∈ E0 and x1 ∈ E1. Then, we can define the following interpolation spaces. The space (E0, E1)ρ,q,K, ρ ∈ B and 0 < q ≤ +∞, consists of all x ∈ E0 + E1 which norm n −1 n ||x||ρ,q;K = ||( ρ(2 ) K(2 ,x; E) )n∈Z||ℓq (Z) is finite. The space (E0, E1)ρ,q;J , consists of all x ∈ E0 + E1, which it has a ∞ representation x = n=−∞ un where (un) ∈ E0 ∩ E1 and converges in E + E , which norm 0 1 P n −1 n ||x||ρ,q;J = inf ||( ρ(2 ) J(2 , un; E) )n∈Z||ℓq (Z), is finite, where the infimum is taken over all representations x = un. Besides, we have for the interpolation space (E , E ) that if x ∈ E ∩E 0 1 ρ,q,J P0 1 then ||x|| ||x|| ≤ C||x|| ρ 1 . E 0 ||x|| 0 5 For 0 <q ≤ +∞, the Equivalence Theorem between the J and K method holds, that is, (E0, E1)ρ,q;J = (E0, E1)ρ,q;K. 2.5. The spaces of class Jρ and Kρ. Let E be an intermediate space +− respect to a Banach couple E = (E0, E1) and ρ ∈ B . We say that E is an intermediate space of class Jρ(E0, E1) if the following embedding holds ,E0, E1)ρ,1;J ֒→ E) (9) and we say that E is an intermediate space of class Kρ(E0, E1) if the fol- lowing embedding holds .E ֒→ (E0, E1)ρ,∞;K (10) We note that E is of class Jθ(E0, E1) if and only if for all x ∈ E0 ∩ E1, it holds ||x|| (11) ||x|| ≤ C||x|| ρ E1 .
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