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

Abstract It turned out in recent papers that the fractional maximal operator $$M_{\gamma ,s,\alpha }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>α</mml:mi> </mml:mrow> </mml:msub> </mml:math> has an important role in harmonic analysis. In this article, we prove that, under some conditions, $$M_{\gamma ,s,\alpha }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>α</mml:mi> </mml:mrow> </mml:msub> </mml:math> is bounded from variable martingale Hardy-Lorentz spaces to variable Lorentz-Karamata spaces. As an application, the boundedness of the classical fractional maximal operator $$M_\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>α</mml:mi> </mml:msub> </mml:math> and $$M_{\gamma ,s,\alpha }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>α</mml:mi> </mml:mrow> </mml:msub> </mml:math> on variable Lorentz-Karamata spaces are also discussed. Our results are new even for the classical fractional maximal operator.