arXiv 1602.048092016-02math.AP

临界Hardy不等式

Critical Hardy inequalities

Michael Ruzhansky, Durvudkhan Suragan

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

摘要：

证明了幂零李群的一个最一般的子类，即齐次群类上的一系列临界Hardy不等式和不确定性原理。此外，我们建立了一类新的临界Hardy不等式，并证明了Hardy-Sobolev型不等式。对于$\mathbb R^{n}$的情况，所获得的大多数估计已经是新的。例如，对于任何$f\in C_{0}^{\infty}(\mathbb{R}^{n}\backslash\{0\})$，我们的结果意味着范围内的形式$$\qquad \underset{R>0}{\sup}\left\|\frac{f-f_{R}}{|x|^{\frac{n}{p}}{\log}\frac{R}{|x|}}\right\|_{L^{p}(\mathbb{R}^{n})}\leq \frac{p}{p-1}\left\| \frac{1}{|x|^{\frac{n}{p}-1}} \nabla f\right\|_{L^{p}(\mathbb{R}^{n})},\quad 1<p<\infty,$$，其中$f_{R}=f(R\frac{x}{|x|})$，尖锐的常数$\frac{p}{p-1}$，恢复已知的情况下，$p=n$和$p=2$的关键哈代不等式。此外，我们的结果还意味着一种新的临界Hardy不等式的形式$$\left\|\frac{f}{|x|}\right\|_{L^{n}(\mathbb{R}^{n})}\leq n\left\|(\log|x|)\nabla f\right\|_{L^{n}\mathbb{R}^{n})}, $$的所有$f\in C_{0}^{\infty}(\mathbb{R}^{n}\backslash\{0\}),$，其中常数$n$是尖锐的。然而，齐次群提供了一个完美的一般性程度来讨论这样的估计，而不使用$\mathbb R^n$或欧几里得距离的特定属性。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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