Product kernels adapted to curves in the space

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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.