Real Interpolation and Two Variants of Gehring's Lemma

Let Ω be a fixed open cube in ℝ n . For r ∈[1, ∞) and α∈[0, ∞) we define( M r ; α , Ω f ) ( x ) = supQ x ∈ Q ⊂ Ω‖ f χ Q ‖L r‖ χ Q ‖L r( log L ) α, x ∈ Ωwhere Q is a cube in ℝ n (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 1.1( M p ; 0 , Ω f ) ( x ) ⩽ c ( M 1 ; 0 , Ω f ) ( x ) , x ∈ Ωfor some p ∈(1, ∞) and c ∈(0, ∞), then there exist q ∈( p , ∞) and C = C ( p , q , n , c )∈(0, ∞) such that( 1 | Q |∫ Q| f ( x ) |q d x )1 / q⩽ c 1 | Q |∫ Q| f ( x ) | d xfor all cubes Q ⊂Ω, where ∣ Q ∣ denotes the n ‐dimensional Lebesgue measure of Q . In particular, a function f ∈ L 1 (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.