On an invariant on isometric immersions into spaces of constant sectional curvature

In the present paper, we give an invariant on isometric immersions into spaces of constant sectional curvature. This invariant is a direct consequence of the Gauss equation and the Codazzi equation of isometric immersions. We apply this invariant on some examples. Further, we apply it to codimension 1 local isometric immersions of 2-step nilpotent Lie groups with arbitrary leftinvariant Riemannian metric into spaces of constant nonpositive sectional curvature. We also consider the more general class, namely, three-dimensional Lie groups G with nontrivial center and with arbitrary left-invariant metric. We show that if the metric of G is not symmetric, then there are no local isometric immersions of G into Q _c ⁴.     

Functional Calculus for Infinitesimal Generators of Holomorphic Semigroups

We give a functional calculus formula for infinitesimal generators of holomorphic semigroups of operators on Banach spaces, which involves the Bochner–Riesz kernels of such generators. The rate of smoothness of operating functions is related to the exponent of the growth on vertical lines of the operator norm of the semigroup. The strength of the formula is tested on Poisson and Gauss semigroups inL¹(Rⁿ) andL¹(G), for a stratified Lie groupG. We give also a self-contained theory of smooth absolutely continuous functions on the half line [0, ∞).     

Spaces of functions of fractional smoothness on an irregular domain

On an irregular domain G ⊂ ℝ ⁿ of a certain type, we introduce spaces of functions of fractional smoothness s > 0. We prove embedding theorems relating these spaces to the Sobolev spaces W _p ^m (G) and Lebesgue spaces L _p (G).     

Characterizations of some classes ofL 1-preduals by the Alfsen-Effros structure topology

Using the Alfsen-Effros structure topology on the extreme boundary of the dual unit ball of a complex Banach space, we give characterizations ofL ¹-preduals (i.e., Banach spaces whose duals are isometrically isomorphic toL ¹ (μ) for a non-negative measure μ) and some of its subclasses viz.G-spaces,C _σ-spaces and _c ⁰(Г) spaces.     

Homogeneous spaces with polar isotropy

 We consider homogeneous spaces G/K with G a simple compact Lie group, endowed with an arbitrary G-invariant Riemannian metric. We classify those spaces where the action of K on G/K is polar and show that such spaces are locally symmetric. Moreover we give a classification of pairs (G,K) with G compact semisimple such that K has polar linear isotropy representation. Received: 16 May 2002 / Revised version: 8 November 2002 Published online: 3 March 2003 Mathematics Subject Classification (2000): 53C35, 57S15     

p-Power Points and Modules of Constant p-Power Jordan Type

We study finitely generated modules over k[G] for a finite abelian p-group G, char(k) = p, through restrictions to certain subalgebras of k[G]. We define p-power points, shifted cyclic p-power order subgroups of k[G], and give characterizations of these. We define modules of constant p ^t -Jordan type, constant p ^t -power-Jordan type as generalizations of modules of constant Jordan type, and p ^t -support, nonmaximal p ^t -support spaces. We obtain a filtration of modules of constant Jordan type with modules of constant p-power Jordan type as the last term and give examples of non-isomorphic modules of constant p-power Jordan type having the same constant Jordan type.     

Function spaces of Lizorkin-Triebel type on an irregular domain

On an irregular domain G ⊂ ℝ ⁿ of a certain type, we introduce function spaces of fractional smoothness s > 0 that are similar to the Lizorkin-Triebel spaces. We prove embedding theorems that show how these spaces are related to the Sobolev and Lebesgue spaces W _p ^m (G) and L _p (G). Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 260, pp. 32–43.     

Bers-orlicz spaces on the product riemann surface

In this paper, the Bers-Orlicz spaces on the automorphic formA _α ^ϕ (G) (orEA _α ^ϕ (G)) andL _α ^ϕ (G) on the product Riemann surfaces are studied. We prove that eachf∈A _α ^ϕ (G) is a cusp form. Forf∈A _α ^ϕ (G), we give the reproducing formula. And, we give the projective operatorP _gga fromL _α ^ϕ (G) toA _α ^ϕ (G) toEA _α ^ϕ (G)). After giving some fundamental properties of the Poincaré series, we prove a dual theoremA _α ^ϕ (G)=(EA _α ^ϕ (G)). Supported by the National Nature Science Foundation of China     

Notes on the minimal number of ramified primes in some l-extensions of Q

For a finite l-group G, let ram ^t (G) denote the minimal integer such that G can be realized as the Galois group of a tamely ramified extension of Q ramified only at ram ^t (G) finite primes. We study the upper bound of ram ^t (G) and give an improvement of the result of Plans. We also give the best bound of ram ^t (G) for all 3-groups G of order less than or equal to 3⁵. Received: 14 September 2007     

Isomorphisms preserving invariants

Let V and W be finite dimensional real vector spaces and let G ì GL(V){G \subset {\rm GL}(V)} and H ì GL(W){H \subset {\rm GL}(W)} be finite subgroups. Assume for simplicity that the actions contain no reflections. Let Y and Z denote the real algebraic varieties corresponding to \mathbbR[V]^G{\mathbb{R}[V]^G} and \mathbbR[W]^H{\mathbb{R}[W]^H}, respectively. If V and W are quasi-isomorphic, i.e., if there is a linear isomorphism L : V → W such that L sends G-orbits to H-orbits and L ⁻¹ sends H-orbits to G-orbits, then L induces an isomorphism of Y and Z. Conversely, suppose that f : Y → Z is a germ of a diffeomorphism sending the origin of Y to the origin of Z. Then we show that V and W are quasi-isomorphic, This result is closely related to a theorem of Strub [8], for which we give a new proof. We also give a new proof of a result of Kriegl et al. [3] on lifting of biholomorphisms of quotient spaces.     

Hardy spaces for non-compactly causal symmetric spaces and the most continuous spectrum

Let G/H be a semisimple symmetric space. Then the space L²(G/H) can be decomposed into a finite sum of series of representations induced from parabolic subgroups of G. The most continuous part of the spectrum of L²(G/H) is the part induced from the smallest possible parabolic subgroup. In this paper we introduce Hardy spaces canonically related to this part of the spectrum for a class of non-compactly causal symmetric spaces. The Hardy space is a reproducing Hilbert space of holomorphic functions on a bounded symmetric domain of tube type, containing G/H as a boundary component. A boundary value map is constructed and we show that it induces a G-isomorphism onto a multiplicity free subspace of full spectrum in the most continuous part L_mc²(G/H) of L²(G/H). We also relate our Hardy space to the classical Hardy space on the bounded symmetric domain.Supported in part by NSF-grant DMS-0070816 and the MSRISupported in part by NSF-grant DMS-0097314 and the MSRISupported in part by NSF-grant DMS-0070607 and the MSRI     

Fourier multipliers and spectral measures in Banach function spaces

It is classical that amongst all spaces L^p (G), 1 ≤ p ≤ ∞, for , or say, only L² (G) (that is, p = 2) has the property that every bounded Borel function on the dual group Γ determines a bounded Fourier multiplier operator in L² (G). Stone’s theorem asserts that there exists a regular, projection-valued measure (of operators on L² (G)), defined on the Borel sets of Γ, with Fourier-Stieltjes transform equal to the group of translation operators on L² (G); this fails for every p ≠ 2. We show that this special status of L² (G) amongst the spaces L^p (G), 1 ≤ p ≤ ∞, is actually more widespread; it continues to hold in a much larger class of Banach function spaces defined over G (relative to Haar measure).      

Spaces in which every dense subset is a Gδ

A topological space X is called a DG_δ-space if every subset of X is a G_δ-set in its closure. In this paper we study DG_δ-spaces that contains subspaces in which every dense subset is open and spaces in which every subset is a G_δ . We give some new results in these classes of topological spaces.     

Symmetric cohomology of groups in low dimension

We give an explicit characterization for group extensions that correspond to elements of the symmetric cohomology HS ²(G, A). We also give conditions for the map HS ⁿ (G, A) → H ⁿ (G, A) to be injective.     

Covering numbers for Chevalley groups

LetG be a quasisimple Chevalley group. We give an upper bound for the covering number cn(G) which is linear in the rank ofG, i.e. we give a constantd such that for every noncentral conjugacy classC ofG we haveC ^rd =G, wherer=rankG. Research supported in part by NSERC Canada Grant A7251. Research supported in part by the Hermann Minkowski-Minerva Center for Geometry at Tel Aviv University.     

Finite p-Groups All of Whose Minimal Nonnormal Subgroups Posses Large Normal Closures

We obtained some results about finite p-groups G with G/H^G being abelian for all nonnormal subgroups H, where H^G denotes the normal closure of H. Moreover, we give a classification of finite p-groups G with G/H^G being cyclic for all nonnormal subgroups H.     

Boundedness of commutators on homogeneous Morrey-Herz spaces over locally compact Vilenkin groups

This paper studies some boundedness results of commutators on a class of new spaces M K˙pα,,qλ (G) named as homogenous Morrey-Herz spaces over locally compact Vilenkin groups.     

The Langlands classification for non-connectedp-adic groups

We give the Langlands classification for a non-connected reductive quasisplitp-adic groupG, under the assumption thatG/G ⁰ is abelian (here,G ⁰ denotes the connected component of the identity ofG). The Langlands classification for non-connected groups is an extension of the Langlands classification from the connected case.     

Harmonic maps into loop spaces of compact Lie groups

We study harmonic maps from Riemann surfaces M to the loop spaces ΩG of compact Lie groups G, using the twistor approach. Harmonic maps into loop spaces are of special interest because of their relation to the Yang-Mills equations on ℝ⁴. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 16, Differential and Functional Differential Equations. Part 2, 2006.