The Boundedness of the Fractional Maximal Operator on Variable Lorentz-Karamata Spaces

It turned out in recent papers that the fractional maximal operator Mγ ,s,α M γ , s , α has an important role in harmonic analysis. In this article, we prove that, under some conditions, Mγ ,s,α M γ , s , α is bounded from variable martingale Hardy-Lorentz spaces to variable Lorentz-Karamata spaces. As an application, the boundedness of the classical fractional maximal operator M_α M α and Mγ ,s,α M γ , s , α on variable Lorentz-Karamata spaces are also discussed. Our results are new even for the classical fractional maximal operator.