Theory of Bessel potentials. IV. Potentials on subcartesian spaces with singularities of polyhedral type
Annales de l'Institut Fourier, Tome 25 (1975) no. 3-4, p. 27-69
Dans les articles précédents sur les potentiels besseliens l’article présent était annoncé comme traitant des variétés avec singularités. Cette dernière notion est mieux définie dans le cadre plus général des espaces sous-cartésiens. Dans un tel espace $X$ nous définissons les potentiels locaux d’ordre réduit $\alpha :u\in {P}_{\mathrm{loc}}^{〈\alpha 〉}\left(X\right)$, si pour toute carte $\left(U,\varphi ,{\mathbf{R}}^{n}\right)$ de la structure de $X,u\circ {\gamma }^{-1}$ peut être étendue de $\varphi \left(U\right)$ à ${\mathbf{R}}^{n}$ entier comme potentiel dans ${P}_{\mathrm{loc}}^{\alpha +\left(n/2\right)}\left({\mathbf{R}}^{n}\right)$. Cette définition n’est pas intrinsèque. On obtient une caractérisation intrinsèque de ${P}_{\mathrm{loc}}^{〈\alpha 〉}\left(X\right)$ quand $X$ est à singularités de type polyhédral, c’est-à-dire quand pour un atlas de $X$ l’image de chaque carte est un ensemble polyhédral (l’union d’un nombre fini des polyhèdres géométriques qui peuvent être de dimensions différentes). Cette caractérisation est donnée par des conditions de compatibilité entre les restrictions de la fonction donnée sur $X$ à certaines variétés composant $X$. Pour définir ces conditions complètement on introduit et étudie la notion restriction abstraite d’une fonction $u\in {P}^{〈-k/2〉}\left({\mathbf{R}}^{n}\right)={P}^{\left(n-k\right)/2}\left({\mathbf{R}}^{n}\right)$ à ${\mathbf{R}}^{k}$, $k.
In the previous parts of the series on Bessel potentials the present part was announced as dealing with manifolds with singularities. The last notion is best defined in the more general framework of subcartesian spaces. In a subcartesian space $X$ we define the local potentials of reduced order $\alpha :u\in {P}_{\mathrm{loc}}^{〈\alpha 〉}\left(X\right)$, if for any chart $\left(U,\varphi ,{\mathbf{R}}^{n}\right)$ of the structure of $X,u\circ {\gamma }^{-1}$ can be extended from $\varphi \left(U\right)$ to the whole of ${\mathbf{R}}^{n}$ as potential in ${P}_{\mathrm{loc}}^{\alpha +\left(n/2\right)}\left({\mathbf{R}}^{n}\right)$. This definition is not intrinsic. We obtain an intrinsic characterization of ${P}_{\mathrm{loc}}^{〈\alpha 〉}\left(X\right)$ when $X$ is with singularities of polyhedral type, i.e. form some atlas of $X$ the image of each chart is a polyhedral set (finite union of geometric polyhedra, possibly of different dimensions). This characterization is given in terms of compatibility conditions between the restrictions of the given function $u$ on $X$ to certain manifolds composing $X$. In order to define a complete set of compatibility conditions we introduce and investigate the notion of abstract restriction of a function $u\in {P}^{〈-k/2〉}\left({\mathbf{R}}^{n}\right)={P}^{\left(n-k\right)/2}\left({\mathbf{R}}^{n}\right)$ to ${\mathbf{R}}^{k}$, $k.
@article{AIF_1975__25_3-4_27_0,
author = {Aronszajn, Nachman and Szeptycki, Pawel},
title = {Theory of Bessel potentials. IV. Potentials on subcartesian spaces with singularities of polyhedral type},
journal = {Annales de l'Institut Fourier},
publisher = {Imprimerie Durand},
address = {28 - Luisant},
volume = {25},
number = {3-4},
year = {1975},
pages = {27-69},
doi = {10.5802/aif.572},
zbl = {0304.31010},
mrnumber = {55 \#8780},
language = {en},
url = {http://www.numdam.org/item/AIF_1975__25_3-4_27_0}
}

Aronszajn, Nachman; Szeptycki, Pawel. Theory of Bessel potentials. IV. Potentials on subcartesian spaces with singularities of polyhedral type. Annales de l'Institut Fourier, Tome 25 (1975) no. 3-4, pp. 27-69. doi : 10.5802/aif.572. http://www.numdam.org/item/AIF_1975__25_3-4_27_0/

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