Two-Weight Weak Type Inequality for Modified Fractional Maximal Function Defined with Respect to Non-Doubling Measure and Some Applications in Non-standard Lebesgue Spaces

Abstract

Let $$ \mu $$ a nonnegative Radon measure on $$ {\mathbb{R}}^{d} $$ ; $$ p,q,\gamma ,k $$ real numbers; $$ M_{\mu ,k}^{\gamma } $$ a fractional maximal operator; $$ A_{p,q}^{\gamma ,k} \left( \mu \right) $$ a Muckenhoupt condition associated to $$ \mu $$ ; $$ L^{p( \cdot )} ({\mathbb{R}}^{d} , \mu ) $$ and $$ F(q, p,\alpha ,\mu )({\mathbb{R}}^{d} ) $$ two generalized Lebesgue spaces. The purpose of the present work is double: