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A Maximal Description for the Real Interpolation Method in the Quasi-Banach Case.

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omslaga 1..6 {orders}ms/000562/bergh.3d -3.10.00 - 07:46 A MAXIMAL DESCRIPTION FOR THE REAL INTERPOLATION METHOD IN THE QUASI-BANACH CASE. JO« RAN BERGH and FERNANDO COBOS1 Abstract We give a maximal description in the sense of Aronszajn-Gagliardo for the real method in the category of quasi-Banach spaces. Let A A0;A1 be a couple of quasi-Banach spaces, 0 < q � 1 and 0 < � < 1. The real interpolation space A�;q A0;A1 �;q consists of all a 2 A0 A1 which have a finite quasi-norm kak�;q X1 m ÿ1 2ÿ�mK 2m; a ÿ �q !1=q if 0 < q <1 kak�;q sup m2Z 2ÿ�mK 2m; a � if q 1 where K t; : is the K-functional of Peetre, defined by K t; a inf a0k kA0 t a1k kA1 : a a0 a1; ai 2 Ai n o : As an example, let us recall that Lp0 ;Lp1 �;q Lq (equivalent quasi-norms) provided 0 < p0; p1; q � 1; 0 < � < 1 and 1q 1ÿ�p0 �p1. We refer to the book by Bergh and Lo« fstro« m [2] for full details on the real method. The space A0;A1 �;q is a quasi-Banach space, but if A0;A1 is a Banach couple and 1 � q � 1, then A0;A1 �;q turns out to be a Banach space. Be- sides, working with Banach spaces, the real interpolation method can be de- scribed as a maximal functor in the sense of Aronszajn and Gagliardo [1]. Next we review that description. 1� Supported in part by DGICYT 9PB94-0252) Received November 3, 1997. MATH. SCAND. 87 (2000), 22^26 {orders}ms/000562/bergh.3d -3.10.00 - 07:46 Given two Banach couples A A0;A1 ; B B0;B1 , we write T 2l A;B to mean that T is a linear operator from A0 A1 into B0 B1 whose restric- tion to each Ai (i=0,1) defines a bounded operator from Ai into Bi. Set Tk kA;B max Tk kA0;B0 ; Tk kA1;B1 n o Operators ending in the couple ‘1 ‘1; ‘1 2ÿm will be of special in- terest for us. Scalar sequence spaces are defined over Z and given any se- quence of positive numbers wm , we put ‘q wm �m : �m k k‘q wm wm�m k k‘q<1 n o : Janson proved in [4] that the real method in the category of Banach spaces coincides

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