Operator algebras on manifolds with isolated singularities.

*(English. Russian original)*Zbl 1181.35358
Differ. Equ. 39, No. 1, 99-111 (2003); translation from Differ. Uravn. 39, No. 1, 92-104 (2003).

From the introduction: Modern elliptic theory on manifolds with isolated singularities deals with singularities of two types, conical and cuspidal (see the figure). From the analytic viewpoint, the type of a singularity is determined by the local structure ring of differential operators [B.-W. Schulze, B. Sternin and V. Shatalov, Structure rings of singularities and differential equations. Differential equations, asymptotic analysis, and mathematical physics. Berlin: Akademie Verlag. Math. Res. 100, 325–347 (1997; Zbl 0907.58067) and Differential equations on singular manifolds. Semiclassical theory and operator algebras, Berlin (1998)], and the corresponding sets of pseudodifferential operators (\(\Psi\)DO) on such manifolds are constructed as some extensions of the structure ring. It is interesting (and quite unexpectable) that the resulting set of \(\Psi\)DO on manifolds with cuspidal singularities is an algebra, while \(\Psi\)DO on manifolds with conical singularities do not form an algebra. Therefore, the conical case is exceptional in the theory of equations on manifolds with singularities. The present paper explains this fact.

##### MSC:

35S05 | Pseudodifferential operators as generalizations of partial differential operators |

58J40 | Pseudodifferential and Fourier integral operators on manifolds |

47L80 | Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.) |

47G30 | Pseudodifferential operators |