Product kernels adapted to curves in the space

Abstract

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.