Let Ω be a fixed open cube in ℝn.
For r∈[1, ∞) and α∈[0, ∞)
we define
formula here
where Q is a cube in ℝn
(with sides parallel to the coordinate axes) and χQ
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 Lq(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.