### Product kernels adapted to curves in the space

Valentina Casarino , Paolo Ciatti , and Silvia Secco
Source: Rev. Mat. Iberoamericana Volume 27, Number 3 (2011), 1023-1057.

#### 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.

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Primary Subjects: 42B20, 44A35