On a dual property of the maximal operator on weighted variable L^{p} spaces
نویسندگان
چکیده
where the supremum is taken over all cubes Q ⊂ R containing the point x. In [5], L. Diening proved the following remarkable result: if p− > 1, p+ < ∞ and M is bounded on Lp(·), then M is bounded on L (·), where p′(x) = p(x) p(x)−1 . Despite its apparent simplicity, the proof in [5] is rather long and involved. In this paper we extend Diening’s theorem to weighted variable Lebesgue spaces L p(·) w equipped with norm ‖f‖ L p(·) w = ‖fw‖Lp(·) . We assume that a weight w here is a non-negative function such that w(·)p(·) and w(·)−p(·) are locally integrable. The spaces L w have been studied in numerous works; we refer to the monographs [3,6] for a detailed bibliography.
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