Theory of Bessel potentials. IV. Potentials on subcartesian spaces with singularities of polyhedral type
Annales de l'Institut Fourier, Volume 25 (1975) no. 3-4, p. 27-69

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.

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.

@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},
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, Volume 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/

[1] N. Aronszajn and K. T. Smith, Theory of Bessel Potentials I, Ann. Inst. Fourier, 11 (1961), 385-475. | Numdam | MR 26 #1485 | Zbl 0102.32401

[2] R. D. Adams, N. Aronszajn and K. T. Smith, Theory of Bessel Potentials II, Ann. Inst. Fourier, 17, 2 (1967), 1-135. | Numdam | MR 37 #4281 | Zbl 0185.19703

[3] R. D. Adams, N. Aronszajn and M. S. Hanna, Theory of Bessel Potentials III, Potentials on regular manifolds, Ann. Inst. Fourier, 19, 2 (1969), 279-338. | Numdam | MR 54 #915 | Zbl 0176.09902

[4] N. Aronszajn, Some integral inequalities, Proceedings of the Symposium on Inequalities at Colorado Springs, 1967. | Zbl 0226.26018

[5] N. Aronszajn and G. H. Hardy, Properties of a class of double integrals, Ann. of Math., 46 (1945), 220-241, Errata, ibid. 47 (1946), 166. | MR 7,116b | Zbl 0060.14202

[6] N. Aronszajn and P. Szeptycki, Subcartesian spaces, in preparation. | Zbl 0451.58006

[7] N. Aronszajn, R. D. Brown and R. S. Butcher, Construction of the solution of boundary value problems for the biharmonic operator in a rectangle, Ann. Inst. Fourier, 23, 3 (1973), 49-89. | Numdam | MR 50 #760 | Zbl 0258.31009

[8] Hardy-Littlewood-Polya, Inequalities, Cambridge, 1959.

[9] P. Szeptycki, On restrictions of functions in the spaces Pα,p and Bα,p, Proceedings AMS, 16, 3 (1965), 341-347. | MR 32 #2609 | Zbl 0125.34306

[10] P. Szeptycki, Extensions by mollifiers in Besov spaces, to appear in Studia Mathematica. | Zbl 0338.46030

[11] Ch. D. Marshall, DeRham Cohomology of Subcartesian Structures, Technical Report 24 (new series), University of Kansas, 1971.

[12] K. Spallek, Differenzierbare Räume, Math. Ann., 180 (1969), 269-296. | MR 41 #5655 | Zbl 0169.52901

[13] K. Spallek, Glattung differenzierbarer Räume, Math. Ann., 186 (1970), 233-248. | MR 41 #4566 | Zbl 0184.25001

[14] J. L. Lions and E. Magenes, Non-homogenous boundary value problems and applications, Springer-Verlag, 1972.

[15] P. Grisvard, Equations différentielles abstraites, Ann. Ec. Norm. Sup., Paris (4), 2 (1969). | Numdam | MR 42 #5101 | Zbl 0193.43502