International Journal of Mathematics and Mathematical Sciences
Volume 27 (2001), Issue 8, Pages 495-503
doi:10.1155/S0161171201006548
Rough Marcinkiewicz integral operators
Department of Mathematics, Yarmouk University, Irbid, Jordan
Received 16 January 2001
Copyright © 2001 Hussain Al-Qassem and Ahmad Al-Salman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We study the Marcinkiewicz integral operator M𝒫f(x)=(∫−∞∞|∫|y|≤2tf(x−𝒫(y))(Ω(y)/|y|n−1)dy|2dt/22t)1/2, where 𝒫 is a polynomial mapping from ℝn into ℝd and Ω is a homogeneous function of degree zero on ℝn with mean value zero over the unit sphere Sn−1. We prove an Lp boundedness result of M𝒫 for rough Ω.