Abstract We establish in this paper general geometric Hardy's identities and inequalities on domains in R N in the spirit of their celebrated works by Brezis-Vazquez and Brezis-Marcus. Hardy's identities are powerful tools in establishing more precise and significantly stronger inequalities than those Hardy's inequalities in the literature. More precisely, we use the notion of Bessel pairs introduced by Ghoussoub and Moradifam to establish several Hardy identities and inequalities with the general distance functions. Our distance functions can be understood as the distance to the surfaces of codimension α ∈ R , and include the distance to the origin ( α = N ), the distance to the boundary ( α = 1 ), and even the distance to surfaces of codimension k ∈ N with 1 ≤ k ≤ N , as special cases. Our Hardy's identities for general Bessel pairs and their special cases improve the Hardy inequalities with general distance functions d ( x ) by Barbatis, Filippas and Tertikas. We also establish the best constant and extremal functions for a new type of the Hardy-Sobolev-Maz'ya inequality on the domain Σ = { x i > 0 , i = 1 , . . . , N } with the distance function d 1 ( x ) = d ( x , ∂ Σ ) = min ⁡ { x 1 , . . . , x N } . Our Hardy's identities on the domain { 0 d 1 ( x ) R } with the distance function d 1 ( x ) = min ⁡ { x 1 , . . . , x N } are in the spirit of Brezis-Vazquez and Brezis-Marcus. More applications of our main theorems on Hardy's identities and inequalities with general distance functions are given and these results improve and sharpen many Hardy's inequalities in the literature. Interesting examples of Bessel pairs and distance functions are given as well.

中文翻译：

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具有一般距离函数的几何哈代不等式

摘要 我们在本文中根据 Brezis-Vazquez 和 Brezis-Marcus 的著名著作的精神，在 RN 的域上建立了一般几何 Hardy 的恒等式和不等式。哈代恒等式是建立比文献中的哈代不等式更精确和明显更强的不等式的有力工具。更准确地说，我们使用 Ghoussoub 和 Moradifam 引入的 Bessel 对的概念来建立几个具有一般距离函数的 Hardy 恒等式和不等式。我们的距离函数可以理解为到codimension α ∈ R 曲面的距离，包括到原点的距离( α = N )、到边界的距离( α = 1 )，甚至到codimension 曲面的距离k ∈ N 1 ≤ k ≤ N ，作为特殊情况。我们的哈迪 一般贝塞尔对及其特殊情况的 s 恒等式通过 Barbatis、Filippas 和 Tertikas 改进了具有一般距离函数 d ( x ) 的 Hardy 不等式。我们还为域 Σ = { xi > 0 , i = 1 , 上的新型 Hardy-Sobolev-Maz'ya 不等式建立了最佳常数和极值函数。. . , N } 与距离函数 d 1 ( x ) = d ( x , ∂ Σ ) = min ⁡ { x 1 , . . . , x N } 。我们的哈代在域 { 0 d 1 ( x ) R } 上的恒等式与距离函数 d 1 ( x ) = min ⁡ { x 1 , . . . , x N } 符合 Brezis-Vazquez 和 Brezis-Marcus 的精神。给出了我们的主要定理在具有一般距离函数的哈代恒等式和不等式上的更多应用，这些结果改进和锐化了文献中的许多哈代不等式。