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

Abstract

Abstract We show that if the Hardy–Littewood maximal operator M is bounded on a reflexive variable exponent space L p ⁢ ( ⋅ ) ⁢ ( ℝ d ) {L^{p(\,\cdot\,)}(\mathbb{R}^{d})} , then for every q ∈ ( 1 , ∞ ) {q\in(1,\infty)} , the exponent p ⁢ ( ⋅ ) {p(\,\cdot\,)} admits, for all sufficiently small θ > 0 {\theta>0} , the representation 1 p ⁢ ( x ) = θ q + 1 - θ r ⁢ ( x ) {\frac{1}{p(x)}=\frac{\theta}{q}+\frac{1-\theta}{r(x)}} , x ∈ ℝ d {x\in\mathbb{R}^{d}} , such that the operator M is bounded on the variable Lebesgue space L r ⁢ ( ⋅ ) ⁢ ( ℝ d ) {L^{r(\,\cdot\,)}(\mathbb{R}^{d})} . 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.