Fractional type Marcinkiewicz integral operators on function spaces

Abstract

Abstract In this paper, we discussed about the boundedness of the fractional type Marcinkiewicz integral operators, and improved a result given by Chen, Fan and Ying in 2002. They showed that under certain conditions the fractional type Marcinkiewicz integral operators are bounded from the Triebel–Lizorkin spaces <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mover accent="true"> <m:mi>F</m:mi> <m:mo>˙</m:mo> </m:mover> <m:mrow> <m:mi>p</m:mi> <m:mi>q</m:mi> </m:mrow> <m:mi>α</m:mi> </m:msubsup> <m:mrow> <m:mo>(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> ${\dot{F}_{pq}^{\alpha }({\mathbb {R}^n})}$ to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> ${L^p({\mathbb {R}^n})}$ . We greatly weakened their assumptions and extended their results to a pretty much more general way.