Real interpolation and compactness
Fernando Cobos · Revista Matemática Complutense · 1989
The behavicur of compactness under real interpolation it diseusted.Classical results due to Krasnosel'skii, Lions-Peetre, Persson and Hayakawa are described, as weIl as others obtained very recently by Edmunds, Potter, Fernández and Ihe author.lb we itave (for example) an integral operatorusually tite function space witere tite operator is defined is nol uniquely establisited by given conditions.Obten investigated is tite operator acting between several function spaces.For titjs reason, it is important to itave results witicit give relationsitips between properties ofa given operator considered in two different spaces.Tite celebrated Riesz-Titorin titeorem is a non-trivial example of sucit a resulí.Leí us recaí!its slatement.La (Q, 1i) be a measure space, witit ji a positive measure, and Iet L0 (1 =p = oc) denote tite space of alí (equivalence classes of) p-measurable functionsfon ~, sucit titat ¡1/Ib, = Is finite.Riesz-Thorin Theorem.Assunie thai p0 # p, q, != eralor w/zich niaps L0 coniinuously mío L~(1 = íinuouslv mío L4, wirerep andqaregiven by q,, and leí T he a linear op-0.1),tiren T maps L0 conhp = (I-O)¡p0 + OIp,, hg = <1-O)/q, + O/q, <O < O < 1) 1980 Mathemat¡cs Suhject Clastification (1985 revision): 46M35.47H05