On interpolation of reflexive variable Lebesgue spaces on which the Hardy-Littlewood maximal operator is bounded

We show that if the Hardy-Littewood maximal operator M is bounded on a reflexive variable exponent space Lp(·) (ℝd), then for every q ϵ (1, ∞), the exponent p(·) admits, for all sufficiently small θ > 0, the representation 1/p(x) = θ/q + 1 - θ/ r(x), x ϵ ℝd, such that the operator M is bounded on the variable Lebesgue space Lr(·) (ℝd). This result can be applied for transferring properties like compactness of linear operators from standard Lebesgue spaces to variable Lebesgue spaces by using interpolation techniques.