We establish -boundedness for a class of operators that are given by convolution with product kernels adapted to curves in the space. The 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 -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 estimates for analytic families of fractional operators along curves in the space.
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Valentina Casarino, Paolo Ciatti, Silvia Secco, Product kernels adapted to curves in the space. Rev. Mat. Iberoam. 27 (2011), no. 3, pp. 1023–1057DOI 10.4171/RMI/662