On Complementary Spaces of the Lizorkin Spaces

Let S(Rⁿ){\cal S}(R^n) be the Schwartz space on R ⁿ . For a subspace V ì S(Rⁿ)V\subset {\cal S}(R^n), if a subspace W ì S(Rⁿ)W \subset {\cal S}(R^n) satisfies the condition that S(Rⁿ){\cal S}(R^n) is a direct sum of V and W, then W is called a complementary space of V in S(Rⁿ){\cal S}(R^n). In this article we give complementary spaces of two kinds of the Lizorkin spaces in S(Rⁿ){\cal S}(R^n).     

Localic Krull dimension

We shall define localic Krull dimension for topological spaces. In particular, a space X has the localic Krull dimension n if n is the greatest number such that X can be mapped, via a continuous and open map, onto the n-chain seen as an Alexandroff space. We shall discuss the applications of this concept in obtaining topological completeness results in modal logic. We shall also show how the localic Krull dimension is related to the Krull dimension in ring theory. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Methoden zur Bestimmung von Reprokernen

Methods to determine reproducing kernels. The explicit representation of continuous linear functionals on a Hilbert space by reprokernels is significant for interpolation and approximation. Starting with the kernels theorem, due to Schwartz, we develop methods to determine reprokernels for the Sobolev spaces W₂^k(Ω) if Ω R¹, and for some subspaces of W₂^k(Ω) if ΩRⁿ. Then we determine reprokernels for tensor products of Hilbert spaces. In addition to this we consider three types of limits of reprokernels.     

Algebraic K-theory of non-linear projective spaces

In the spirit of “The Fundamental Theorem for the algebraic K-theory of spaces: I” (J. Pure Appl. Algebra 160 (2001) 21–52) we introduce a category of sheaves of topological spaces on n-dimensional projective space and present a calculation of its K-theory, a “non-linear” analogue of Quillen's isomorphism K_i(P_Rⁿ)₀ⁿK_i(R).     

On resolvable primal spaces

A topological space is called resolvable if it is a union of two disjoint dense subsets, and is n-resolvable if it is a union of n mutually disjoint dense subsets. Clearly a resolvable space has no isolated points. If f is a selfmap on X, the sets A?X with f (A)?A are the closed sets of an Alexandroff topology called the primal topology 𝒫(f ) associated with f. We investigate resolvability for primal spaces (X, 𝒫(f)). Our main result is that an Alexandroff space is resolvable if and only if it has no isolated points. Moreover, n-resolvability and other related concepts are investigated for primal spaces.     

Hyperbolic rank and subexponential corank of metric spaces

We introduce a new quasi-isometry invariant corank X of a metric space X called subexponential corank. A metric space X has subexponential corank k if roughly speaking there exists a continuous map , T is a topological space, such that for each the set g ^-1(t) has subexponential growth rate in X and the topological dimension dimT = k is minimal among all such maps. Our main result is the inequality for a large class of metric spaces X including all locally compact Hadamard spaces, where rank _h X is the maximal topological dimension of among all CAT(—1) spaces Y quasi-isometrically embedded into X (the notion introduced by M. Gromov in a slightly stronger form). This proves several properties of rank _h conjectured by Gromov, in particular, that any Riemannian symmetric space X of noncompact type possesses no quasi-isometric embedding of the standard hyperbolic space H ⁿ with . Submitted: February 2001, Revised: October 2001.     

Ons-sets and mutual absolute continuity of measures on homogeneous spaces

We extend a recent result of A. Jonsson about mutual absolute continuity of twoD _s -measures on ans-setF ⊂R ⁿ to the homogeneous spaces (X, d, μ) of Coifman, Weiss. Here we define Hausdorff measure, Hausdorff dimension,D _s -set andd-set relative to the measureμ. Our main result holds for so called (s, d)-sets,d ≥s, and is stronger than Jonssons result even inR ⁿ . As applications we interpret this Hausdorff dimension as a relative dimension for very regular sets and show that it in general depends strongly onμ. For this purpose we construct a strictly increasing functionf :R →R, whose measure is doubling and concentrated on a set of arbitrary small Hausdorff dimension. The extension off to a quasiconformal map of the half plane onto itself sharpens a classical example of Ahlfors-Beurling.     

W2m(Rn)中单个函数的平移和 伸缩组合的稠密性

本文讨论单个函数的平移和伸缩的组合在Sobolev空间W     

The Embedding W n,1 into C 0 on Compact Riemannian Manifolds

We study the best constant in the inequality corresponding to the Sobolev embedding W ^n,1(R ⁿ ) into the space of bounded continuous functions C ₀(R ⁿ ). Then, we adapt this inequality on compact Riemannian manifolds and discuss on its optimality.     

Strong capacity-estimates for “fractional” norms

It is proved that for all fractionall the integral \(\int\limits_0^\infty {(p,\ell ) - cap(M_t )} dt^p\) is majorized by the P-th power norm of the functionu in the space ? _p ^l (Rⁿ) (here M_t={x∶¦u(x)¦?t} and (p,l)-cap(e) is the (p,l)-capacity of the compactum e?Rⁿ). Similar results are obtained for the spaces W _p ^l (Rⁿ) and the spaces of M. Riesz and Bessel potentials. One considers consequences regarding imbedding theorems of “fractional” spaces in ?_q(dμ), whereμ is a nonnegative measure in Rⁿ. One considers specially the case p=1.     

Spectral convergence for vibrating systems containing a part with negligible mass

We consider a set of Neumann (mixed, respectively) eigenvalue problems for the Laplace operator. Each problem is posed in a bounded domain Ω_R of ?ⁿ, with n=2,3, which contains a fixed bounded domain B where the density takes the value 1 and 0 outside. Ω_R has a diameter depending on a parameter R, with R?1, diam(Ω_R) →∞ as R→∞ and the union of these sets is the whole space ?ⁿ (the half space {x∈?ⁿ/x_n<0}, respectively). Depending on the dimension of the space n, and on the boundary conditions, we describe the asymptotic behaviour of the eigenelements as R→∞. We apply these asymptotics in order to derive important spectral properties for vibrating systems with concentrated masses. Copyright © 2005 John Wiley & Sons, Ltd.     

On Homotopy Types of Alexandroff Spaces

We generalise some results of R. E. Stong concerning finite spaces to wider subclasses of Alexandroff spaces. In particular, we characterize pairs of spaces X,Y such that the compact-open topology on C(X,Y) is Alexandroff, give a homotopy type classification of a class of infinite Alexandroff spaces and prove some results concerning cores of locally finite spaces. We also discuss a mistake found in an article of F.G. Arenas. Since the category of T ₀ Alexandroff spaces is equivalent to the category of posets, our results may lead to a deeper understanding of the notion of a core of an infinite poset.     

On some properties of the functions from Sobolev-Morrey type spaces

In this paper the spaces of type Sobolev-Morrey-W _p,a,г,τ ^l (Q,G)-are constructed, the differential properties are studied and it is proved that the functions from these spaces satisfy Holder's condition, in the case, if the domain G∋R ⁿ satisfies the flexible λ-horn condition.     

Groups of Isometries of Hardy-Type Spaces of Analytic Functionals in the Unit Ball of the Space~L p

The tensor structure of spaces L _p (R ⁿ ) of summable functions of several variables is described. A scale of Hardy-type spaces of analytic functionals defined in the unit ball of the space L _p (R ¹) of summable functions of one variable is introduced. One-parameter groups of isometries of such spaces of analytic functionals are investigated.     

Equidistribution of expanding measures with local maximal dimension and diophantine approximation

We consider improvements of Dirichlet’s Theorem on the space of matrices M_m,n(\mathbb R){M_{m,n}(\mathbb R)}. It is shown that for a certain class of fractals K ì [0,1]^mn ì M_m,n(\mathbb R){K\subset [0,1]^{mn}\subset M_{m,n}(\mathbb R)} of local maximal dimension Dirichlet’s Theorem cannot be improved almost everywhere. This is shown using entropy and dynamics on homogeneous spaces of Lie groups.     

Embedding of Stein spaces into sequence spaces

In this note we show that a connected, reduced Stein space X of arbitrary dimension admits a holomorphic embedding into various sequence spaces, for example into s,s',0(ⁿ) or 1,T₂,...,T_n>, and also into infinite dimensional complex Banach spaces. As an application we prove that the Fréchet space 0 (X) of holomorphic functions on X is a quotient of s.     

On the euclidean dimension of a wheel

Following Erdös, Harary, and Tutte, the euclidean dimension of a graphG is the minimumn such thatG can be embedded in euclideann-spaceR ⁿ so that each edge ofG has length 1. We present constructive proofs which give the euclidean dimension of a wheel and of a complete tripartite graph. We also define the generalized wheelW _m,n as the join and determine the euclidean dimension of all generalized wheels.     

Stability problems in a theorem of F. Schur

Schur's theorem states that an isotropic Riemannian manifold of dimension greater than two has constant curvature. It is natural to guess that compact almost isotropic Riemannian manifolds of dimension greater than two are close to spaces of almost constant curvature. We take the curvature anisotropy as the discrepancy of the sectional curvatures at a point. The main result of this paper is that Riemannian manifolds in Cheeger's class ℜ(n,d,V,A) withL ₁-small integral anisotropy haveL _p-small change of the sectional curvature over the manifold. We also estimate the deviation of the metric tensor from that of constant curvature in theW _p ² -norm, and prove that compact almost isotropic spaces inherit the differential structure of a space form. These stability results are based on the generalization of Schur' theorem to metric spaces.     

Embedding theorems for Bergman spaces in quasiconformal balls

The authors prove some embedding theorems for Bergman type spaces of functions defined on quasiconformal balls inR ⁿ,n≥2. This article was processed by the author using the Springer-Verlag T_EX mamath marco package 1990     

On the Elementary Retarded,Ultrahyperbolic Solution of the Klein-Gordon Operator,Iterated k Times

Let t = (t₁,…,t_n) be a point of ?ⁿ. We shall write . We put, by the definition, W_α(u, m) = (m^?2u)^{(α ? n)/4}[π^{(n ? 2)/2}2^{(α + n ? 2)/2}Г(α/2)]J_{(α ? n)/2}(m²u)^1/2; here α is a complex parameter, m a real nonnegative number, and n the dimension of the space. W_α(u, m), which is an ordinary function if Re α ≥ n, is an entire distributional function of α. First we evaluate {□ + m²}W_{α + 2}(u, m) = W_α(u, m), where {□ + m²} is the ultrahyperbolic operator. Then we express W_α(u, m) as a linear combination of R_α(u) of differntial orders; R_α(u) is Marcel Riesz's ultrahyperbolic kernel. We also obtain the following results: W_?2k(u, m) = {□ + m²}^kδ, k = 0, 1,…; W₀(u, m) = δ; and {□ + m²}^kW_2k(u, m) = δ. Finally we prove that W_α(u, m = 0) = R_α(u). Several of these results, in the particular case µ = 1, were proved earlier by a completely different method.