Nonhomogeneous central Morrey-type spaces in Lp(⋅) and weak estimates for the maximal and Riesz potential operators

Abstract

Our aim in this paper is to discuss the weak estimate for the maximal and potential operators in the nonhomogeneous central Morrey-type space $M^{p(\cdot ),q,\mathrm{\nu }}({\mathbf{R}}^n)$, which consists of all measurable functions f on ${\mathbf{R}}^n$ satisfying

$$\Vert f{\Vert }_{M^{p(\cdot ),q,\mathrm{\nu }}({\mathbf{R}}^n)}={\left({\int }_1^{\mathrm{\infty }}{(r^{-\mathrm{\nu }}\Vert f{\Vert }_{L^{p(\cdot )}(B(0,r))})}^q\frac{dr}{r}\right)}^{1∕q}<\mathrm{\infty },$$

where $\mathrm{\nu }>0$, $0<q<\mathrm{\infty }$ and $p(\cdot )$ is a variable exponent satisfying the log-Hölder condition at infinity. When $q=\mathrm{\infty }$, we apply a necessary modification.