Natural differential operators and graph complexes

We show how the machine invented by S. Merkulov [S.A. Merkulov, Operads, deformation theory and F-manifolds, in: Frobenius Manifolds, Aspects Math., vol. E36, Vieweg, Wiesbaden, 2004, pp. 213-251; S.A. Merkulov, PROP profile of deformation quantization, Preprint, math.QA/0412257, December 2004; S.A. Merkulov, PROP profile of Poisson geometry, Comm. Math. Phys. 262 (1) (February 2006) 117-135] can be used to study and classify natural operators in differential geometry. We also give an interpretation of graph complexes arising in this context in terms of representation theory. As application, we prove several results on classification of natural operators acting on vector fields and connections.     

Weak type estimates for maximal operators over convex hypersurfaces

We prove sharp weak type (p,p) estimates on H ^p spaces for the maximal operators with a rough distance function over convex hypersurfaces.     

Opial-type inequalities for differential operators

Some continuous and discrete versions of Opial-type inequalities which are readily applicable to differential and difference operators are established. These generalize earlier results of Anastassiou and Pe?ari?, and of Koliha and Pe?ari?.     

Deformations of hypersurfaces in equiaffine differential geometry

We study infinitesimal normal deformations in equiaffine hypersurface theory and prove local and global rigidity results. Main tool are uniqueness theorems for certain p.d.e. of second order.     

Commuting differential operators

We consider a model problem, related to Bessel functions, in the theory of commutative rings of linear differential operators with one independent variable. We construct new series of such commutative rings and give simple examples of rings with three generators.     

Covariant differential operators

Supported in part by the Louisiana Education Quality Support Fund 86-LBR-016-04     

The Beilinson equivalence for differential operators

Let D be the ring of differential operators on a smooth irreducible affine variety X over C, or, more generally, the enveloping algebra of any locally free Lie algebroid on X. The category of finitely generated graded modules of the Rees algebra has a natural quotient category P_D which imitates the category of modules on Proj of a graded commutative ring. We show that the derived category D^b(P_D) is equivalent to the derived category of finitely generated modules of a sheaf of algebras E on X which is coherent over X. This generalizes the usual Beilinson equivalence for projective space, and also the Beilinson equivalence for differential operators on a smooth curve used by Ben-Zvi and Nevins in [6] to describe the moduli space of left ideals in D.     

Evolution equations for abstract differential operators

We study in this paper the wellposedness and regularity of solutions of evolution equations associated with abstract differential operators on a Banach space. The results can be applied to many partial differential equations on different function spaces.     

Bounds for eigenvalues of singular differential operators

The spectrum of the Sturm-Liouville operator is studied, and the principal results consist of two-sided bounds of the eigenvalue distribution and of a criterion establishing that the resolvent belongs to a symmetrically normed ideal .Translated from Matematicheskie Zametki, Vol. 20, No. 6, pp. 859–867, December, 1976.In conclusion the author expresses his gratitude to M. Z. Solomyak for valuable critical remarks.