Tohoku Mathematical Journal 2007 June SECOND SERIES VOL. 59, NO. 2

 Tohoku Math. J. 59 (2007), 145-166

Title A ROUGH MULTIPLE MARCINKIEWICZ INTEGRAL ALONG CONTINUOUS SURFACES

Author Huoxiong Wu

(Received May 14, 2004, revised December 27, 2006)
Abstract. By means of the method of block decompositions for kernel functions and some delicate estimates on Fourier transforms, the $L^p(\boldsymbol{R}^m \times \boldsymbol{R}^n \times \boldsymbol{R})$-boundedness of the multiple Marcinkiewicz integral is established along a continuous surface with rough kernel for some $p>1$. The condition on the integral kernel is the best possible for the $L^2$-boundedness of the multiple Marcinkiewicz integral operator.

2000 Mathematics Subject Classification. Primary 42B20; Secondary 42B25, 42B99.

Key words and phrases. Marcinkiewicz integral, continuous surface, rough kernel, block spaces, product spaces, Fourier transform estimate, Littlewood-Paley theory.