Approximation by M.~Riesz's kernels in $L^p$ for $p1$

A. B. Aleksandrov

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 32, Tome 315 (2004), pp. 5-38

Résumé

Let $\alpha>0$. We consider the linear span ${\mathfrak X}_\alpha(\mathbb R^n)$ of scalar Riesz's kernels $\{\frac1{|x-a|^\alpha}\}_{a\in\mathbb R^n}$ and the linear span ${\mathfrak Y}_\alpha(\mathbb R^n)$ of vector Riesz's kernels $\{\frac1{|x-a|^{\alpha+1}}(x-a)\}_{a\in\mathbb R^n}$. We deal with the following questions. 1. When is the intersection ${\mathfrak X}_\alpha(\mathbb R^n)\cap L^p(\mathbb R^n)$ dense in $L^p(\mathbb R^n)$? 2. When is the intersection ${\mathfrak Y}_\alpha(\mathbb R^n)\cap L^p(\mathbb R^n,\mathbb R^n)$ dense in $L^p(\mathbb R^n,\mathbb R^n)$?