## Tohoku Mathematical Journal 2009 March SECOND SERIES VOL. 61, NO. 1

 Tohoku Math. J. 61 (2009), 41-65

Title ON THE BOUNDEDNESS OF SINGULAR INTEGRALS WITH VARIABLE KERNELS

Author Qingying Xue and Kôzô Yabuta

(Received July 19, 2007, revised June 23, 2008)
Abstract. We prove the $L^p (1<p<\infty)$ estimates for the singular integrals with rough variable kernels. The $L^p$ boundedness of a class of modified directional Hilbert transforms is also given. As a consequence of this result, we get a good estimate for the singular integrals with rough odd kernels.

2000 Mathematics Subject Classification. Primary 42B25; Secondary 47G10.

Key words and phrases. Singular integrals, variable kernels, $L^p$ estimates, $L^\infty\times L^q(S^{n-1})$ spaces.