Spherical homogeneous spaces of minimal rank

Let G be a connected reductive algebraic group over an algebraically closed field K of characteristic zero. Let G/B denote the complete flag variety of G. A G-homogeneous space G/H is said to be spherical if H has finitely many orbits in G/B. A class of spherical homogeneous spaces containing the tori, the complete homogeneous spaces and the group G (viewed as a G×G-homogeneous space) has particularly nice properties. Namely, the pair (G,H) is called a spherical pair of minimal rank if there exists x in G/B such that the orbit H.x of x by H is open in G/B and the stabilizer Hx of x in H contains a maximal torus of H. In this article, we study and classify the spherical pairs of minimal rank.     

On a generalization of distance sets

A subset X in the d-dimensional Euclidean space is called a k-distance set if there are exactly k distinct distances between two distinct points in X and a subset X is called a locally k-distance set if for any point x in X, there are at most k distinct distances between x and other points in X.Delsarte, Goethals, and Seidel gave the Fisher type upper bound for the cardinalities of k-distance sets on a sphere in 1977. In the same way, we are able to give the same bound for locally k-distance sets on a sphere. In the first part of this paper, we prove that if X is a locally k-distance set attaining the Fisher type upper bound, then determining a weight function w, (X,w) is a tight weighted spherical 2k-design. This result implies that locally k-distance sets attaining the Fisher type upper bound are k-distance sets. In the second part, we give a new absolute bound for the cardinalities of k-distance sets on a sphere. This upper bound is useful for k-distance sets for which the linear programming bound is not applicable. In the third part, we discuss about locally two-distance sets in Euclidean spaces. We give an upper bound for the cardinalities of locally two-distance sets in Euclidean spaces. Moreover, we prove that the existence of a spherical two-distance set in (d−1)-space which attains the Fisher type upper bound is equivalent to the existence of a locally two-distance set but not a two-distance set in d-space with more than d(d+1)/2 points. We also classify optimal (largest possible) locally two-distance sets for dimensions less than eight. In addition, we determine the maximum cardinalities of locally two-distance sets on a sphere for dimensions less than forty.     

Bounds for solid angles of lattices of rank three

We find sharp absolute constants C₁ and C₂ with the following property: every well-rounded lattice of rank 3 in a Euclidean space has a minimal basis so that the solid angle spanned by these basis vectors lies in the interval [C₁,C₂]. In fact, we show that these absolute bounds hold for a larger class of lattices than just well-rounded, and the upper bound holds for all. We state a technical condition on the lattice that may prevent it from satisfying the absolute lower bound on the solid angle, in which case we derive a lower bound in terms of the ratios of successive minima of the lattice. We use this result to show that among all spherical triangles on the unit sphere in RN with vertices on the minimal vectors of a lattice, the smallest possible area is achieved by a configuration of minimal vectors of the (normalized) face centered cubic lattice in R³. Such spherical configurations come up in connection with the kissing number problem.     

Strongly hyperbolically convex functions

Let C(w₁,w₂,w₃) denote the circle in through w₁,w₂,w₃ and let denote one of the two arcs between w₁,w₂ belonging to C(w₁,w₂,w₃). We prove that a domain Ω in the Riemann sphere, with no antipodal points, is spherically convex if and only if for any w₁,w₂,w₃∈Ω, with w₁≠w₂, the arc of the circle which does not contain lies in Ω. Based on this characterization we call a domain G in the unit disk D, strongly hyperbolically convex if for any w₁,w₂,w₃∈G, with w₁≠w₂, the arc in D of the circle is also contained in G. A number of results on conformal maps onto strongly hyperbolically convex domains are obtained.     

Smooth hypersurfaces with a CR contraction

In this paper we show that a C^∞ real hypersurface in Cn+1 of finite D'Angelo type admitting a weakly contracting local CR automorphism is CR equivalent to a weighted homogeneous hypersurface. As an application, we show that a bounded pseudoconvex domain in Cn+1 with C^∞ boundary of finite D'Angelo type with a hyperbolic orbit accumulation point is biholomorphically equivalent to a domain defined by a weighted homogeneous polynomial.     

Explicit Hilbert spaces for certain unipotent representations III

In this paper we construct a family of small unitary representations for real semisimple Lie groups associated with Jordan algebras. These representations are realized on L²-spaces of certain orbits in the Jordan algebra. The representations are spherical and one of our key results is a precise L²-estimate for the Fourier transform of the spherical vector. We also consider the tensor products of these representations and describe their decomposition.     

Spherical functions on Euclidean space

We study spherical functions on Euclidean spaces from the viewpoint of Riemannian symmetric spaces. Here the Euclidean space En=G/K where G is the semidirect product Rn⋅K of the translation group with a closed subgroup K of the orthogonal group O(n). We give exact parameterizations of the space of (G,K)—spherical functions by a certain affine algebraic variety, and of the positive definite ones by a real form of that variety. We give exact formulae for the spherical functions in the case where K is transitive on the unit sphere in En.     

Absolute continuity of convolutions of orbital measures on Riemannian symmetric spaces

We study the absolute continuity of the measures and of on the Riemannian symmetric spaces X of noncompact type for nonzero elements Xj, X∈a. For m,l?r+1, where r is the rank of X, the considered convolutions have a density. We conjecture that the condition m,l?r+1 is necessary. The conjecture is proved for the symmetric spaces of type An−1. Moreover, the minimal value of l is determined, in function of the irregularity of X.     

Some weighted estimates for Littlewood-Paley functions and radial multipliers

We prove some weighted estimates for certain Littlewood-Paley operators on the weighted Hardy spaces H_w^p (0<p?1) and on the weighted L^p spaces. We also prove some weighted estimates for the Bochner-Riesz operators and the spherical means.     

Contraction obstructions for treewidth

We provide two parameterized graphs Γk, Πk with the following property: for every positive integer k, there is a constant ck such that every graph G with treewidth at least ck, contains one of Kk, Γk, Πk as a contraction, where Kk is a complete graph on k vertices. These three parameterized graphs can be seen as “obstruction patterns” for the treewidth with respect to the contraction partial ordering. We also present some refinements of this result along with their algorithmic consequences.     

On the essential commutant of analytic Toeplitz operators associated with spherical isometries

Let T∈Bn(H) be an essentially normal spherical isometry with empty point spectrum on a separable complex Hilbert space H, and let AT⊂B(H) be the unital dual operator algebra generated by T. In this note we show that every operator S∈B(H) in the essential commutant of AT has the form S=X+K with a T-Toeplitz operator X and a compact operator K. Our proof actually covers a larger class of subnormal operator tuples, called A-isometries, which includes for example the tuple T=(Mz₁,…,Mzn)∈B(H²n(σ)) consisting of the multiplication operators with the coordinate functions on the Hardy space H²(σ) associated with the normalized surface measure σ on the boundary ∂D of a strictly pseudoconvex domain D⊂Cn. As an application we determine the essential commutant of the set of all analytic Toeplitz operators on H²(σ) and thus extend results proved by Davidson (1977) [6] for the unit disc and Ding and Sun (1997) [11] for the unit ball.     

Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, II

In [S.G. Samko, B.G. Vakulov, Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, J. Math. Anal. Appl. 310 (2005) 229-246], Sobolev-type p(⋅)→q(⋅)-theorems were proved for the Riesz potential operator Iα in the weighted Lebesgue generalized spaces Lp(⋅)(Rn,ρ) with the variable exponent p(x) and a two-parameter power weight fixed to an arbitrary finite point x₀ and to infinity, under an additional condition relating the weight exponents at x₀ and at infinity. We show in this note that those theorems are valid without this additional condition. Similar theorems for a spherical analogue of the Riesz potential operator in the corresponding weighted spaces Lp(⋅)(Sn,ρ) on the unit sphere Sn in Rn+1 are also improved in the same way.     

Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators

We prove Sobolev-type p(⋅)→q(⋅)-theorems for the Riesz potential operator Iα in the weighted Lebesgue generalized spaces Lp(⋅)(Rn,ρ) with the variable exponent p(x) and a two-parametrical power weight fixed to an arbitrary finite point and to infinity, as well as similar theorems for a spherical analogue of the Riesz potential operator in the corresponding weighted spaces Lp(⋅)(Sn,ρ) on the unit sphere Sn in Rn+1.     

Heckman-Opdam's Jacobi polynomials for the BCn root system and generalized spherical functions

We prove that the radial part of the Laplacian on the space of generalized spherical functions on the symmetric space GL(m+n)/GL(m)×GL(n) is the Sutherland differential operator for the root system BC_n and the radial parts of the differential operators corresponding to the higher Casimirs yield the integrals of the quantum Calogero-Moser system. It allows us to give a representation theoretical construction for the three parameter family of Heckman-Opdam's Jacobi polynomials for the BC_n root system.     

Blowup phenomena of solutions to the Euler equations for compressible fluid flow

The blowup phenomena of solutions is investigated for the Euler equations of compressible fluid flow. The approach is to construct special explicit solutions with spherical symmetry to study certain blowup behavior of multi-dimensional solutions. In particular, the special solutions with velocity of the form c(t)x are constructed to show the expanding and blowup properties. The solution with velocity of the form for γ?1 and for any space dimensions is obtained as a corollary. Another conclusion is that there is only trivial solution with velocity of the form c(t)|x|^α-1x for α≠1 and multi-space dimensions.     

Dilations of Contraction Cocycles and Cocycle Perturbations of the Translation Group of the Line

The class of contraction cocycles which can be dilated to unitary Markovian cocycles of a translation group S on the straight line is introduced. The class of cocycle perturbations of S by unitary Markovian cocycles W with the property W _t ? I ∈ S ₂ (the Hilbert—Schmidt class) is investigated. The results are applied to perturbations of Kolmogorov flows on hyperfinite factors generated by the algebra of canonical anticommutation relations.     

Symmetry of large solutions of nonlinear elliptic equations in a ball

Let g be a locally Lipschitz continuous real-valued function which satisfies the Keller-Osserman condition and is convex at infinity, then any large solution of −Δu+g(u)=0 in a ball is radially symmetric.     

Iterative schemes for computing fixed points of nonexpansive mappings in Banach spaces

Let X be a real Banach space with a normalized duality mapping uniformly norm-to-weak^? continuous on bounded sets or a reflexive Banach space which admits a weakly continuous duality mapping JΦ with gauge ?. Let f be an α-contraction and {Tn} a sequence of nonexpansive mappings, we study the strong convergence of explicit iterative schemes

(1)     

Bounds for the sum of distances of spherical sets of small size

We derive upper and lower bounds on the sum of distances of a spherical code of size N in n dimensions when $N=\mathrm{\Theta }(n^{\alpha }),0<\alpha ?2$. The bounds are derived by specializing recent general, universal bounds on energy of spherical sets. We discuss asymptotic behavior of our bounds along with several examples of codes whose sum of distances closely follows the upper bound.