Critical Hardy inequalities (1602.04809v3)

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 $$\qquad \underset{R>0}{\sup}\left|\frac{f-f_{R}}{|x|^{\frac{n}{p}}{\log}\frac{R}{|x|}}\right|_{L^{p}(\mathbb{R}^{n})}\leq \frac{p}{p-1}\left| \frac{1}{|x|^{\frac{n}{p}-1}} \nabla f\right|{L^{p}(\mathbb{R}^{n})},\quad 1<p<\infty,$$ 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 $$\left|\frac{f}{|x|}\right|{L^{n}(\mathbb{R}^{n})}\leq n\left|(\log|x|)\nabla f\right|{L^{n}\mathbb{R}^{n})}, $$ 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.