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

Nachman Aronszajn; Pawel Szeptycki

Annales de l'institut Fourier (1975)

- Volume: 25, Issue: 3-4, page 27-69
- ISSN: 0373-0956

## Access Full Article

top## Abstract

top## How to cite

topAronszajn, 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 $\{\bf R\}^k$, $k< n$.},

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 ${\bf R}^k$, $k< n$.

LA - eng

UR - http://eudml.org/doc/74247

ER -

## References

top- [1] N. ARONSZAJN and K. T. SMITH, Theory of Bessel Potentials I, Ann. Inst. Fourier, 11 (1961), 385-475. Zbl0102.32401MR26 #1485
- [2] R. D. ADAMS, N. ARONSZAJN and K. T. SMITH, Theory of Bessel Potentials II, Ann. Inst. Fourier, 17, 2 (1967), 1-135. Zbl0185.19703MR37 #4281
- [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. Zbl0176.09902MR54 #915
- [4] N. ARONSZAJN, Some integral inequalities, Proceedings of the Symposium on Inequalities at Colorado Springs, 1967. Zbl0226.26018
- [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. Zbl0060.14202MR7,116b
- [6] N. ARONSZAJN and P. SZEPTYCKI, Subcartesian spaces, in preparation. Zbl0451.58006
- [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. Zbl0258.31009MR50 #760
- [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. Zbl0125.34306MR32 #2609
- [10] P. SZEPTYCKI, Extensions by mollifiers in Besov spaces, to appear in Studia Mathematica. Zbl0338.46030
- [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. Zbl0169.52901MR41 #5655
- [13] K. SPALLEK, Glattung differenzierbarer Räume, Math. Ann., 186 (1970), 233-248. Zbl0184.25001MR41 #4566
- [14] J. L. LIONS and E. MAGENES, Non-homogenous boundary value problems and applications, Springer-Verlag, 1972. Zbl0223.35039
- [15] P. GRISVARD, Equations différentielles abstraites, Ann. Ec. Norm. Sup., Paris (4), 2 (1969). Zbl0193.43502MR42 #5101

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.