Critical Hardy inequalities

Abstract

We prove a range of critical Hardy inequalities and uncertainty type principles on one of most general subclasses of nilpotent Lie groups, namely the class of homogeneous groups. Moreover, we establish a new type of critical Hardy inequality and prove Hardy-Sobolev type inequalities. Most of the obtained estimates are new already for the case of ${\mathbb{R}}^n$. For example, for any $f\in C_0^{\infty }({\mathbb{R}}^n\backslash \{0\})$ our results imply the range of critical Hardy inequalities of the form $$\underset{R>0}{\sup }{\Vert \frac{f-f_R}{|x{|}^{\frac{n}{p}}\log \frac{R}{|x|}}\Vert }_{L^p({\mathbb{R}}^n)}\le \frac{p}{p-1}{\Vert \frac{1}{|x{|}^{\frac{n}{p}-1}}\nabla f\Vert }_{L^p({\mathbb{R}}^n)},1<p<\infty ,$$ where $f_R=f(R\frac{x}{|x|})$, with sharp constant $\frac{p}{p-1}$, recovering the known cases of $p=n$ and $p=2$. Moreover, our results also imply a new type of a critical Hardy inequality of the form $${\Vert \frac{f}{|x|}\Vert }_{L^n({\mathbb{R}}^n)}\le n{\Vert (\log |x|)\nabla f\Vert }_{L^n{\mathbb{R}}^n)},$$ for all $f\in C_0^{\infty }({\mathbb{R}}^n\backslash \{0\}),$ where the constant $n$ is sharp. However, homogeneous groups provide a perfect degree of generality to talk about such estimates without using specific properties of ${\mathbb{R}}^n$ or of the Euclidean distance.

19 pages; new inequalities have been added yielding also new results on Rn; therefore, the title has been also changed