Critical Hardy inequalities

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 $f\in C_0^{\mathrm{\infty }}({\mathbb{R}}^n\mathrm{\backslash }\{0\})$ our results imply the range of critical Hardy inequalities of the form $${\parallel \frac{f-f_R}{\mathrm{\mid }x{\mathrm{\mid }}^{\frac{n}{p}}\log \frac{R}{\mathrm{\mid }x\mathrm{\mid }}}\parallel }_{L^p({\mathbb{R}}^n)}\le \frac{p}{p-1}{\parallel \frac{1}{\mathrm{\mid }x{\mathrm{\mid }}^{\frac{n}{p}-1}}\mathrm{\nabla }f\parallel }_{L^p({\mathbb{R}}^n)},1<p<\mathrm{\infty },$$ where $f_R=f(R\frac{x}{\mathrm{\mid }x\mathrm{\mid }})$, 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 $${\parallel \frac{f}{\mathrm{\mid }x\mathrm{\mid }}\parallel }_{L^n({\mathbb{R}}^n)}\le n{\parallel (\log \mathrm{\mid }x\mathrm{\mid })\mathrm{\nabla }f\parallel }_{L^n{\mathbb{R}}^n)},$$ for all $f\in C_{0}^{\infty}($\mathbb{R}$^{n}\backslash{0}),$wheretheconstant$n$ 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.