We establish $L^p$-boundedness for a class of operators that are given by convolution with product kernels adapted to curves in the space. The $L^p$ bounds follow from the decomposition of the adapted kernel into a sum of two kernels with singularities concentrated respectively on a coordinate plane and along the curve. The proof of the $L^p$-estimates for the two corresponding operators involves Fourier analysis techniques and some algebraic tools, namely the Bernstein-Sato polynomials. As an application, we show that these bounds can be exploited in the study of $L^p-L^q$ estimates for analytic families of fractional operators along curves in the space.
"Product kernels adapted to curves in the space." Rev. Mat. Iberoamericana 27 (3) 1023 - 1057, September, 2011.