Revista Matemática Iberoamericana


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Volume 27, Issue 3, 2011, pp. 1023–1057
DOI: 10.4171/RMI/662

Published online: 2011-12-06

Product kernels adapted to curves in the space

Valentina Casarino[1], Paolo Ciatti[2] and Silvia Secco[3]

(1) Università degli Studi di Padova, Vicenza, Italy
(2) Università degli Studi di Padova, Italy
(3) Padova, Italy

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.

Keywords: product kernels, $L^p$ estimates, convolution, Bernstein-Sato polynomials

Casarino Valentina, Ciatti Paolo, Secco Silvia: Product kernels adapted to curves in the space. Rev. Mat. Iberoamericana 27 (2011), 1023-1057. doi: 10.4171/RMI/662