A Note on a Class of Generalized Parabolic Marcinkiewicz Integrals along Surfaces of Revolution

Abstract

In this article, certain sharp Lp estimates for a specific class of generalized Marcinkiewicz operators with mixed homogeneity associated to surfaces of revolution are established. By virtue of Yano’s extrapolation argument, beside these estimates, the Lp boundedness of the aforementioned operators under weaker assumptions on the kernels is confirmed. The obtained results in this article are fundamental extensions and improvements of numerous previously known results on parabolic generalized Marcinkiewicz integrals.

Keywords

• extrapolation
• Triebel–Lizorkin space
• generalized Marcinkiewicz
• rough kernels