The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces Lp(Rd) (in the case p > 1), but (in the case when 1/p(·) is log-Hölder continuous and p- = inf{p(x): x\in Rd > 1) on the variable Lebesgue spaces Lp(·)(Rd), too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type (1, 1). In the present note we generalize Besicovitch’s covering theorem for the so-called γ-rectangles. We introduce a general maximal operator Msγδ, and with the help of generalized Φ-functions, the strong- and weak-type inequalities will be proved for this maximal operator. Namely, if the exponent function 1/p(·) is log-Hölder continuous and p- ≥ s, where 1 ≤ s ≤ ∞ is arbitrary (or
p- ≥ s), then the maximal operator Msγδ is bounded on the space Lp(·)(Rd) (or the maximal operator is of weak-type (p(·), p(·)))., Kristóf Szarvas, Ferenc Weisz., and Obsahuje seznam literatury