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We show that if the Hardy–Littewood maximal operator  M  is bounded on a reflexive variable exponent spaceLp⁢(⋅)⁢(ℝd){L^{p(\,\cdot\,)}(\mathbb{R}^{d})}, then for everyq∈(1,∞){q\in(1,\infty)}, the exponentp⁢(⋅){p(\,\cdot\,)}admits, for all sufficiently smallθ&gt;0{\theta&gt;0}, the representation1p⁢(x)=θq+1-θr⁢(x){\frac{1}{p(x)}=\frac{\theta}{q}+\frac{1-\theta}{r(x)}},x∈ℝd{x\in\mathbb{R}^{d}}, such that the operator  M  is bounded on the variable Lebesgue spaceLr⁢(⋅)⁢(ℝd){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.

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