Theory of Bessel potentials. IV. Potentials on subcartesian spaces with singularities of polyhedral type

Aronszajn, Nachman, and Szeptycki, Pawel. "Theory of Bessel potentials. IV. Potentials on subcartesian spaces with singularities of polyhedral type." Annales de l'institut Fourier 25.3-4 (1975): 27-69. <http://eudml.org/doc/74247>.

@article{Aronszajn1975,
abstract = {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$\alpha : u \in P^\{\langle \alpha \rangle \}_\{\rm loc\}(X)$, if for any chart $(U,\varphi ,\{\bf R\}^n)$ of the structure of $X, u\circ \gamma ^\{-1\}$ can be extended from $\varphi (U)$ to the whole of $\{\bf R\}^n$ as potential in $P^\{\alpha +(n/2)\}_\{\rm loc\}(\{\bf R\}^n)$. This definition is not intrinsic. We obtain an intrinsic characterization of $ P^\{\langle \alpha \rangle \}_\{\rm loc\}(X)$ 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 $u\in P^\{\langle -k/2\rangle \}(\{\bf R\}^n) = P^\{(n-k)/2\}(\{\bf R\}^n)$ to $\{\bf R\}^k$, $k&lt; n$.},
author = {Aronszajn, Nachman, Szeptycki, Pawel},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {3-4},
pages = {27-69},
publisher = {Association des Annales de l'Institut Fourier},
title = {Theory of Bessel potentials. IV. Potentials on subcartesian spaces with singularities of polyhedral type},
url = {http://eudml.org/doc/74247},
volume = {25},
year = {1975},
}

TY - JOUR
AU - Aronszajn, Nachman
AU - Szeptycki, Pawel
TI - Theory of Bessel potentials. IV. Potentials on subcartesian spaces with singularities of polyhedral type
JO - Annales de l'institut Fourier
PY - 1975
PB - Association des Annales de l'Institut Fourier
VL - 25
IS - 3-4
SP - 27
EP - 69
AB - 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$\alpha : u \in P^{\langle \alpha \rangle }_{\rm loc}(X)$, if for any chart $(U,\varphi ,{\bf R}^n)$ of the structure of $X, u\circ \gamma ^{-1}$ can be extended from $\varphi (U)$ to the whole of ${\bf R}^n$ as potential in $P^{\alpha +(n/2)}_{\rm loc}({\bf R}^n)$. This definition is not intrinsic. We obtain an intrinsic characterization of $ P^{\langle \alpha \rangle }_{\rm loc}(X)$ 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 $u\in P^{\langle -k/2\rangle }({\bf R}^n) = P^{(n-k)/2}({\bf R}^n)$ to ${\bf R}^k$, $k&lt; n$.
LA - eng
UR - http://eudml.org/doc/74247
ER -