REAL INTERPOLATION AND TWO VARIANTS OF GEHRING'S LEMMA

Abstract

Let Ω be a fixed open cube in ℝⁿ. For r∈[1, ∞) and α∈[0, ∞) we define

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where Q is a cube in ℝⁿ (with sides parallel to the coordinate axes) and χ_Q stands for the characteristic function of the cube Q.

A well-known result of Gehring [5] states that if

formula here

for some p∈(1, ∞) and c∈(0, ∞), then there exist q∈(p, ∞) and C=C(p, q, n, c)∈(0, ∞) such that

formula here

for all cubes Q⊂Ω, where [mid ]Q[mid ] denotes the n-dimensional Lebesgue measure of Q. In particular, a function f∈L¹(Ω) satisfying (1.1) belongs to L^q(Ω).

In [9] it was shown that Gehring's result is a particular case of a more general principle from the real method of interpolation. Roughly speaking, this principle states that if a certain reversed inequality between K-functionals holds at one point of an interpolation scale, then it holds at other nearby points of this scale. Using an extension of Holmstedt's reiteration formulae of [4] and results of [8] on weighted inequalities for monotone functions, we prove here two variants of this principle involving extrapolation spaces of an ordered pair of (quasi-) Banach spaces. As an application we prove the following Gehring-type lemmas.