Real Interpolation and Two Variants of Gehring's Lemma |
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Authors: | Milman Mario; Opic Bohumir |
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Institution: | Department of Mathematics, Florida Atlantic University Boca Raton, FL 33431, USA. E-mail: milman{at}acc.fau.edu
Mathematical Institute of the Czech Academy of Sciences itná 25, 115 67 Praha 1, Czech Republic. E-mail: opic{at}math.cas.cz |
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Abstract: | Let be a fixed open cube in Rn. For r1, ) and 0, ) we define
where Q is a cube in Rn (with sides parallel to the coordinateaxes) and Q stands for the characteristic function of the cubeQ. A well-known result of Gehring 5] states that if
(1.1) for some p(1, ) and c(0, ), then there exist q(p, ) and C=C(p,q, n, c)(0, ) such that
for all cubes Q, where |Q| denotes the n-dimensional Lebesguemeasure of Q. In particular, a function fL1() satisfying (1.1)belongs to Lq(). In 9] it was shown that Gehring's result is a particular caseof a more general principle from the real method of interpolation.Roughly speaking, this principle states that if a certain reversedinequality between K-functionals holds at one point of an interpolationscale, then it holds at other nearby points of this scale. Usingan extension of Holmstedt's reiteration formulae of 4] andresults of 8] on weighted inequalities for monotone functions,we prove here two variants of this principle involving extrapolationspaces of an ordered pair of (quasi-) Banach spaces. As an applicationwe prove the following Gehring-type lemmas. |
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