Tohoku Mathematical Journal
2010
June
SECOND SERIES VOL. 62, NO. 2

Tohoku Math. J.
62 (2010), 233-262

Title BOUNDEDNESS OF THE MARCINKIEWICZ INTEGRALS WITH ROUGH KERNEL ASSOCIATED TO SURFACES

Author Yong Ding, Qingying Xue and Kôzô Yabuta

(Received April 8, 2009, revised January 4, 2010)
Abstract. In this paper, the authors discuss the weighted $L^p$ boundedness for the rough Marcinkiewicz integrals associated to surfaces. More precisely, the kernel of our operator lacks smoothness not only on the unit sphere, but also in the radial directions. Moreover, the surface is defined by using a differentiable function with monotonicity and some properties on the positive real line. The results given in this paper improve and extend some known results.

2000 Mathematics Subject Classification. Primary 42B25; Secondary 47G10.
Key words and phrases. Marcinkiewicz integrals, $L^p$ boundedness, weighted boundedness, rough kernel.

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