Product kernels adapted to curves in the space

TL;DR

This paper proves $L^p$ bounds for convolution operators with product kernels adapted to curves in space, using kernel decomposition, Fourier analysis, and Bernstein-Sato polynomials.

Contribution

It introduces a novel approach to establish $L^p$-boundedness for such operators by decomposing kernels and applying advanced algebraic and Fourier analysis techniques.

Findings

Established $L^p$-boundedness for a new class of convolution operators

Decomposed kernels into parts with singularities on coordinate planes and curves

Applied Bernstein-Sato polynomials in the analysis

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 sigularities 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.