Bounds for singular integrals associated with a class of hypersurfaces

Abstract

For the hypersurface Γ=(y,γ(y)), the singular integral operator along Γ is defined by.

$$Tf(x,x_n ) = P.V.\int_{\mathbb{R}^n } {, f(x - y,x_n ) - } \gamma (y))_{\left| y \right|^{n - 1} }^{\Omega (v)} dy$$

where Σ is homogeneous of order 0,$\int_{\Sigma _{n \lambda } } {\Omega (y')dy'} = 0$. For a certain class of hypersurfaces, T is shown to be bounded on L^p(Rⁿ) provided Ω∈L ¹_α (Σ _n−2),P>1.