Characterization of weights for the variable fractional maximal operator and weighted inequalities for variable fractional rough operators

We characterize the class of weights related to the boundedness of variable fractional maximal operator $M_{\beta(\cdot),r(\cdot)}$ on variable Lebesgue spaces. This extend previously known results, including those corresponding to the fractional operator $M_{\beta(\cdot),1}$. In addition, we introduce a class of kernels $K$ satisfying a new variable H\"ormander-type condition $H_{\beta(\cdot),r(\cdot)}$. For the fractional operator $T_{\beta(\cdot)}$ given by a kernel in $H_{\beta(\cdot),r(\cdot)}$, we prove a Coifman-Fefferman inequality and weighted inequalities in variable Lebesgue space. Finally, we provide examples of kernels in this variable H\"ormander class.