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arXiv 0903.4624math.AP
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加权Orlicz空间之间Hardy不等式的一个变形

On a variant of Hardy inequality between weighted Orlicz spaces

Agnieszka Kalamajska, Katarzyna Pietruska-Paluba

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34 pages
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摘要：

设M是满足$Δ_2$ -条件的N-函数，$ω, \vp$是另外两个函数，$ω\ge 0$。研究了Hardy型不等式\[ \int_{\rp} M(ω(x)|u(x)|) {\rm exp}(-\vp (x))dx \le C\int_{\rp} M(|u'(x)|) {\rm exp}(-\vp (x))dx, \]，其中$u$属于包含在局部绝对连续函数空间中的某个伸缩不变集${\cal R}$.我们给出了三重$(ω,\vp,M)$必须满足的充分条件，以使这些不等式对给定集合${\cal R}$中的$u$有效。集合${\cal R}$可以小于哈迪变换的集合。还给出了常数的界，从而恢复了经典的Hardy不等式的最佳常数.

英文摘要：

Let M be an N-function satisfying the $Δ_2$- condition, let $ω, \vp$ be two other functions, $ω\ge 0$. We study Hardy-type inequalities \[ \int_{\rp} M(ω(x)|u(x)|) {\rm exp}(-\vp (x))dx \le C\int_{\rp} M(|u'(x)|) {\rm exp}(-\vp (x))dx, \] where $u$ belongs to some dilation invariant set ${\cal R}$ contained in the space of locally absolutely continuous functions. We give sufficient conditions the triple $(ω,\vp,M)$ must satisfy in order to have such inequalities valid for $u$ from a given set ${\cal R}$. The set ${\cal R}$ can be smaller than the set of Hardy transforms. Bounds for constants, retrieving classical Hardy inequalities with best constants, are also given.

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。

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