TY - JOUR

T1 - Weak- and strong-type inequality for the cone-like maximal operator in variable Lebesgue spaces

AU - Szarvas, Kristóf

AU - Weisz, F.

PY - 2016/12/1

Y1 - 2016/12/1

N2 - 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 ∈ 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 Ms γδ, 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 Ms γδ is bounded on the space Lp(·)(Rd) (or the maximal operator is of weak-type (p(·), p(·))).

AB - 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 ∈ 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 Ms γδ, 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 Ms γδ is bounded on the space Lp(·)(Rd) (or the maximal operator is of weak-type (p(·), p(·))).

KW - Besicovitch’s covering theorem

KW - maximal operator

KW - strong-type inequality

KW - variable Lebesgue space

KW - weak-type inequality

KW - γ-rectangle

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U2 - 10.1007/s10587-016-0311-9

DO - 10.1007/s10587-016-0311-9

M3 - Article

AN - SCOPUS:84995685066

VL - 66

SP - 1079

EP - 1101

JO - Czechoslovak Mathematical Journal

JF - Czechoslovak Mathematical Journal

SN - 0011-4642

IS - 4

ER -