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The Engulfing Property for Sections of Convex Functions on the Heisenberg Group and the Associated Quasi-distance
The Journal of Geometric Analysis ( IF 1.2 ) Pub Date : 2021-04-05 , DOI: 10.1007/s12220-021-00648-7
A. Calogero , R. Pini

In this paper we investigate the property of engulfing for H-convex functions defined on the Heisenberg group \({\mathbb {H}^n}\). Starting from the horizontal sections introduced by Capogna and Maldonado (Proc Am Math Soc 134:3191–3199, 2006) , we consider a new notion of section, called \({\mathbb {H}^n}\)-section, as well as a new condition of engulfing associated to the \({\mathbb {H}^n}\)-sections, for an H-convex function defined in \(\mathbb {H}^n.\) These sections, that arise as suitable unions of horizontal sections, are dimensionally larger; as a matter of fact, the \({\mathbb {H}^n}\)-sections, with their engulfing property, will lead to the definition of a quasi-distance in \({\mathbb {H}^n}\) in a way similar to Aimar et al. in the Euclidean case (J Fourier Anal Appl 4:377–381, 1998). A key role is played by the property of round H-sections for an H-convex function, and by its connection with the engulfing properties.



中文翻译:

海森堡群上凸函数的截面的吞没性质及相关的拟距离

在本文中,我们研究了在Heisenberg群\({\ mathbb {H} ^ n} \)上定义的H-凸函数的卷积性质。从Capogna和Maldonado引入的水平截面开始(Proc Am Math Soc 134:3191–3199,2006),我们考虑了一个新的截面概念,称为\({\ mathbb {H} ^ n} \)- section以及与\({\ mathbb {H} ^ n} \)部分相关的吞噬新条件,对于\(\ mathbb {H} ^ n。\)这些部分中定义的H-凸函数,当合适的水平截面结合产生时,尺寸更大;实际上,\({\ mathbb {H} ^ n} \)截面具有吞噬性,将以类似于Aimar等人的方式在\({\ mathbb {H} ^ n} \)中定义准距离。在欧几里得案中(J Fourier Anal Appl 4:377-381,1998)。H凸函数的圆形H形截面的特性及其与吞没特性的联系起着关键作用。

更新日期:2021-04-05
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