Let Ω be a fixed open cube in ℝn.
For r∈[1, ∞) and α∈[0, ∞)
where Q is a cube in ℝn
(with sides parallel to the coordinate axes) and χQ
the characteristic function of the cube Q.
A well-known result of Gehring  states that if
for some p∈(1, ∞) and c∈(0, ∞),
then there exist q∈(p, ∞) and
C=C(p, q, n, c)∈(0,
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  it was shown that Gehring's result is
particular case of a more general
principle from the real method of interpolation. Roughly speaking, this
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 
and results of  on weighted
inequalities for monotone functions, we prove here two variants of this
involving extrapolation spaces of an ordered pair of (quasi-) Banach spaces.
application we prove the following Gehring-type lemmas.