Singular integrals with rough kernels on product spaces
The main result of the paper reads as follows. Suppose that \(\Omega\) is a homogeneous function on \(R^n\times R^m\) of degree zero satisfying \(\int_{S^{n-1}}\Omega(.,y)=\int_{S^{m-1}}\Omega(x,.)=0\), and that \(h\) is a bounded function. Then the operator \[ Tf(x,y)=\text{p.v.}\int h(|\xi|,|\nu|)\Omega(\xi,\nu)|\xi|^{-n} |\nu|^{-m}f(x-\xi,y-\nu)d\xi d\nu \] is bounded in \(L^p(R^n\times R^m)\), provided that \(\Omega\) belongs to a block function space \(B^{0,1}_q(S^{n-1}\times S^{m-1})\) for some \(q>0\). This result extends the results of \textit{J. Duoandikoetxea} [Ann. Inst. Fourier 36, No.~4, 185-206 (1986; Zbl 0596.42007)] and \textit{Y. Jiang} and \textit{S. Lu} [Hokkaido Math. J. 24, No.~1, 1-7 (1995; Zbl 0830.42006)]. Some results on boundedness of the singular integrals of similar type along surfaces complete the paper.
scientific article; zbMATH DE number 1396092