Product kernels adapted to curves in the space

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 Oil a coordinate plane and along the curve. The proof of the L(P)-estimates for the two corresponding operators involves Fourier analysis techniques and sonic 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.