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

Szarvas, Kristóf ; Weisz, Ferenc

Czechoslovak Mathematical Journal, Tome 66 (2016) no. 4, pp. 1079-1101.

Résumé

The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces $L_p({\mathbb{R}}^d)$ (in the case $p>1$), but (in the case when $1/p(\cdot )$ is log-Hölder continuous and $p_{-}=inf\{p(x):x\in {\mathbb{R}}^d\}>1$) on the variable Lebesgue spaces $L_{p(\cdot )}({\mathbb{R}}^d)$, 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 $\gamma$-rectangles. We introduce a general maximal operator $M_s^{\gamma ,\delta }$ and with the help of generalized $\mathrm{\Phi }$-functions, the strong- and weak-type inequalities will be proved for this maximal operator. Namely, if the exponent function $1/p(\cdot )$ is log-Hölder continuous and $p_{-}>s$, where $1\le s\le \mathrm{\infty }$ is arbitrary (or $p_{-}\ge s$), then the maximal operator $M_s^{\gamma ,\delta }$ is bounded on the space $L_{p(\cdot )}({\mathbb{R}}^d)$ (or the maximal operator is of weak-type $(p(\cdot ),p(\cdot ))$).

Détail

DOI : 10.1007/s10587-016-0311-9

Classification : 42B25, 42B35, 52C17
Mots-clés : variable Lebesgue space; maximal operator; $\gamma$-rectangle; Besicovitch's covering theorem; weak-type inequality; strong-type inequality