Weighted criteria for generalized fractional maximal functions and potentials in Lebesgue spaces with variable exponent

Abstract

Necessary and sufficient conditions governing two-weight inequalities with general-type weights for fractional maximal functions and Riesz potentials with variable parameters are established in the Lebesgue spaces with variable exponent. In two-weight inequalities the right-hand side weight to the certain power satisfies the reverse doubling condition. In particular, from the general results we have: generalization of the Sobolev inequality for potentials; criteria governing the trace inequality for fractional maximal functions and potential operators; theorem of Muckenhoupt–Wheeden type (one-weight inequality) for fractional maximal functions defined on a bounded interval when the parameter satisfies the Dini–Lipschitz condition. Sawyer-type two-weight criteria for fractional maximal functions are also derived.

Keywords:

• Lebesgue spaces with variable exponent
• Generalized potentials
• Fractional maximal functions
• Two-weight inequality
• Trace inequality

Acknowledgements

The authors express their gratitude to the referee for helpful remarks.