Product kernels adapted to curves in the space. (English) Zbl 1236.42009

Casarino, Valentina; Ciatti, Paolo; Secco, Silvia

Rev. Mat. Iberoam. 27, No. 3, 1023-1057 (2011).

S. Secco [Math. Z. 248, No. 3, 459-476 (2004; Zbl 1069.47024)] studied the \(L^p\)-boundedness for convolution operators with product kernels adapted to curves in the plane. The purpose of this paper is to extend these results to higher-dimensional spaces.
Denote an element of \({\mathbb R}^3={\mathbb R}\times {\mathbb R}^2\) by the pair \((x_1,x)\), where \(x=(x_2,x_3)\). Assume that \(K_0\) is a product kernel on \({\mathbb R}^3\) and consider the curve \(x=\gamma(x_1)\) with \(\gamma(x_1)=(x_1^m,x_1^n)\), \(x_1\in {\mathbb R}\), \(m,n\in {\mathbb N}\) and \(2\leq m<n\). Define a distribution \(K\) by \[ \int K(x_1,x)f(x_1,x)dx_1dx:= \int K_0(x_1,x)f(x_1,x+\gamma(x_1))dx_1dx \] for a Schwartz function \(f\) on \({\mathbb R}^3\). \(K\) is called an adapted kernel.
The authors prove that the convolution operator \(T: f\mapsto f*K\), initially defined on the Schwartz spaces \({\mathcal S}({\mathbb R}^3)\), extends to a bounded operator on \(L^p({\mathbb R}^3)\) for \(1<p<\infty\). The results still hold for the higher-dimensional spaces \({\mathbb R}^d ={\mathbb R}\times {\mathbb R}^{d-1}\).
The main idea of the proof is to decompose the adapted kernel \(K\) 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, such as the analytic interpolation method, and some algebraic tools, namely the Bernstein-Sato polynomials.
As an application, the authors 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.

Reviewer: Pu Zhang (Mudanjiang)

MSC:

42B20	Singular and oscillatory integrals (Calderón-Zygmund, etc.)
44A35	Convolution as an integral transform

Keywords:

product kernels; \(L^p\) estimates; convolution; Bernstein-Sato polynomials

Citations:

Zbl 1069.47024

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