Classification of topologically conjugate affine mappings

We consider affine mappings from ℝ ⁿ into ℝ ⁿ , n ≥ 1. We prove a theorem on the topological conjugacy of an affine mapping that has at least one fixed point to the corresponding linear mapping. We give a classification, up to topological conjugacy, for affine mappings from R into R and also for affine mappings from ℝ ⁿ into ℝ ⁿ , n > 1, having at least one fixed point and the nonperiodic linear part. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 1, pp. 134–139, January, 2009.     

A notion of geometric complexity and its application to topological rigidity

We introduce a geometric invariant, called finite decomposition complexity (FDC), to study topological rigidity of manifolds. We prove for instance that if the fundamental group of a compact aspherical manifold M has FDC, and if N is homotopy equivalent to M, then M×ℝ ⁿ is homeomorphic to N×ℝ ⁿ , for n large enough. This statement is known as the stable Borel conjecture. On the other hand, we show that the class of FDC groups includes all countable subgroups of GL(n,K), for any field K.     

Positive definite functions and stable random vectors

We say that a random vector X = (X ₁, …, X _n ) in ℝ ⁿ is an n-dimensional version of a random variable Y if, for any a ∈ ℝ ⁿ , the random variables Σa _i X _i and γ(a)Y are identically distributed, where γ: ℝ ⁿ → [0,∞) is called the standard of X. An old problem is to characterize those functions γ that can appear as the standard of an n-dimensional version. In this paper, we prove the conjecture of Lisitsky that every standard must be the norm of a space that embeds in L ₀. This result is almost optimal, as the norm of any finite-dimensional subspace of L _p with p ∈ (0, 2] is the standard of an n-dimensional version (p-stable random vector) by the classical result of P. Lèvy. An equivalent formulation is that if a function of the form f(‖ · ‖ _K ) is positive definite on ℝ ⁿ , where K is an origin symmetric star body in ℝ ⁿ and f: ℝ → ℝ is an even continuous function, then either the space (ℝ ⁿ , ‖·‖ _K ) embeds in L ₀ or f is a constant function. Combined with known facts about embedding in L ₀, this result leads to several generalizations of the solution of Schoenberg’s problem on positive definite functions.     

Smooth convex bodies with proportional projection functions

For a convex body K ⊂ ℝⁿ and i ∈ {1, …, n − 1}, the function assigning to any i-dimensional subspace L of ℝⁿ, the i-dimensional volume of the orthogonal projection of K to L, is called the i-th projection function of K. Let K, K ₀ ⊂ ℝⁿ be smooth convex bodies with boundaries of class C ² and positive Gauss-Kronecker curvature and assume K ₀ is centrally symmetric. Excluding two exceptional cases, (i, j) = (1, n − 1) and (i, j) = (n − 2, n − 1), we prove that K and K ₀ are homothetic if their i-th and j-th projection functions are proportional. When K ₀ is a Euclidean ball this shows that a convex body with C ² boundary and positive Gauss-Kronecker with constant i-th and j-th projection functions is a Euclidean ball. The second author was supported in part by the European Network PHD, FP6 Marie Curie Actions, RTN, Contract MCRN-511953.     

Dini Set-Valued Directional Derivative in Locally Lipschitz Vector Optimization

The present paper studies the following constrained vector optimization problem: min  _C f(x), g(x)∈−K, h(x)=0, where f:ℝ ⁿ →ℝ ^m , g:ℝ ⁿ →ℝ ^p and h:ℝ ⁿ →ℝ ^q are locally Lipschitz functions and C⊂ℝ ^m , K⊂ℝ ^p are closed convex cones. In terms of the Dini set-valued directional derivative, first-order necessary and first-order sufficient conditions are obtained for a point x ⁰ to be a w-minimizer (weakly efficient point) or an i-minimizer (isolated minimizer of order 1). It is shown that, under natural assumptions (given by a nonsmooth variant of the implicit function theorem for the equality constraints), the obtained conditions improve some given by Clarke and Craven. Further comparison is done with some recent results of Khanh, Tuan and of Jiiménez, Novo.     

Multi-parameter singular radon transforms I: The <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> theory

The purpose of this paper is to study the L ² boundedness of operators of the form f ↦ ψ(x) ∫ f (γ _t (x))K(t)dt, where γ _t (x) is a C ^∞ function defined on a neighborhood of the origin in (t, x) ∈ ℝ ^N × ℝ ⁿ , satisfying γ ₀(x) ≡ x, ψ is a C ^∞ cut-off function supported on a small neighborhood of 0 ∈ ℝ ⁿ , and K is a “multi-parameter singular kernel” supported on a small neighborhood of 0 ∈ ℝ ^N . The goal is, given an appropriate class of kernels K, to give conditions on γ such that every operator of the above form is bounded on L ². The case when K is a Calderón-Zygmund kernel was studied by Christ, Nagel, Stein, and Wainger; we generalize their conditions to the case when K has a “multi-parameter” structure. For example, when K is given by a “product kernel.” Even when K is a Calderón- Zygmund kernel, our methods yield some new results. This is the first paper in a three part series, the later two of which are joint with E. M. Stein. The second paper deals with the related question of L ^p boundedness, while the third paper deals with the special case when γ is real analytic.     

On polynomiality of the method of analytic centers for fractional problems

We establish polynomial time convergence of the method of analytic centers for the fractional programming problemt→min |x∈G, tB(x)−A(x)∈K, whereG ⊂ ℝ ⁿ is a closed and bounded convex domain,K ⊂ ℝ ^m is a closed convex cone andA(x):G → ℝ ⁿ ,B(x):G→K are regular enough (say, affine) mappings. This research was partly supported by grant #93-012-499 of the Fundamental Studies Foundation of Russian Academy of Sciences     

Littlewood–Paley theorem on spaces <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">t</Emphasis>)</Superscript>(ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

We point out that if the Hardy–Littlewood maximal operator is bounded on the space L ^p(t)(ℝ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ, then the well-known characterization of the spaces L ^p (ℝ), 1 < p < ∞, by the Littlewood–Paley theory extends to the space L ^p(t)(ℝ). We show that, for n > 1 , the Littlewood–Paley operator is bounded on L ^p(t) (ℝ ⁿ ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ ⁿ , if and only if p(t) = const. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1709–1715, December, 2008.     

On classification of orthogonal multiplications à la do carmo-wallach

The space of range-equivalence classes of full orthogonal multiplications F: ℝ ⁿ ×ℝ ⁿ →ℝ ^p , n≤p ≤n ², is shown to be a compact convex body lying in so(n)⊗so(n). Furthermore, the dimension of the space of equivalence classes is determined to be (n ²(n−1)²)/4−n(n−1).     

Polynomials hardly commuting with increasing bijections

Let I be an interval in the real line ℝ. Among the real polynomials that take I to I, we ask which ones do not commute with any increasing bijection of I other than identity. For this purely algebraic problem, the solution involves concepts in topological dynamics. Our main characterizations are in terms of full orbits of critical points and periodic points. Using these, we obtain simpler criterion, namely, that for no nontrivial subinterval K⊂I, the successive images {f ⁿ (K):n=0,1,2,…} form a pairwise disjoint collection. This problem is of interest in topological dynamics because it is about characterization of polynomials with unique self-topological-conjugacy.     

Extending analyticK-subanalytic functions

Letg:U→ℝ (U open in ℝⁿ) be an analytic and K-subanalytic (i. e. definable in ℝ _an ^K , whereK, the field of exponents, is any subfield ofℝ) function. Then the set of points, denoted Σ, whereg does not admit an analytic extension is K-subanalytic andg can be extended analytically to a neighbourhood of Ū\∑. Partially supported by the European RTN Network RAAG (contract no. HPRN-CT-00271)     

On the Construction of <Emphasis Type="Italic">p</Emphasis>-Harmonic Morphisms and Conformal Actions

We produce p-harmonic morphisms by conformal foliations and Clifford systems. First, we give a useful criterion for a foliation on an m-dimensional Riemannian manifold locally generated by conformal fields to produce p-harmonic morphisms. By using this criterion we manufacture conformal foliations, with codimension not equal to p, which are locally the fibres of p-harmonic morphisms. Then we give a new approach for the construction of p-harmonic morphisms from R^m／{0} to R^n. By the well-known representation of Clifford algebras, we find an abundance of the new 2/3 （m ＋ 1）-harmonic morphism Ф： R^m／{0} → R^n where m = 2κδ（n - 1）.     

Square roots of elliptic second order divergence operators on strongly lipschitz domains:<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> theory

We prove the Kato conjecture for square roots of elliptic second order non-self-adjoint operators in divergence formL = -div(A∇) on strongly Lipschitz domains in ℝⁿ, n≥2, subject to Dirichlet or to Neumann boundary conditions. The method relies on a transference procedure from the recent positive result on ℝⁿ in [2].     

Hardy spaces via distribution spaces

Let ℐ(ℝⁿ) be the Schwartz class on ℝⁿ and ℐ_∞(ℝⁿ) be the collection of functions ϕ ∊ ℐ(ℝⁿ) with additional property that

Approximation by K-finite functions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> spaces

Let Γ ⊂ ℝⁿ, n ≥ 2, be the boundary of a bounded domain. We prove that the translates by elements of Γ of functions which transform according to a fixed irreducible representation of the orthogonal group form a dense class in L ^p (ℝⁿ) for . A similar problem for noncompact symmetric spaces of rank one is also considered. We also study the connection of the above problem with the injectivity sets for weighted spherical mean operators. The first author was supported in part by a grant from UGC via DSA-SAP Phase IV.     

A Paley-Wiener theorem and Wiener-Hopf-type integral equations in Clifford analysis

Using the properties of the monogenic extension of the Fourier transform, we state a Paley-Wiener-type theorem for monogenic functions. Based on an multiplier algebra related to boundary values of monogenic functions we consider integral equations of Wiener-Hopf-typeK±u _±=f on ℝ ⁿ , whereK ∈S′ andu _± are boundary values of monogenic functions in ℝ₊ ⁿ⁺¹ and ℝ_{_} ⁿ⁺¹ respectivly.     

Équivalence rationnelle sur les hypersurfaces cubiques sur les corps <Emphasis Type="Italic">p</Emphasis>-adiques

 We study R-equivalence on cubic hypersurfaces, and explain how to construct families of rational curves. We show that for a smooth cubic hypersurface defined over a number field K, the Chow group of zero-cycles of degree 0 is trivial in almost every place of K. Received: 20 March 2002 Published online: 24 January 2003     

Harnack estimate for the <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">k</Emphasis></Superscript>-flow

We obtain the Harnack estimate of the solution to Hk-flow in Euclidean space Rn 1,for k ＞ 0.By using this estimate,we get some corollaries about the translation soliton.     

On translation and dilation invariant subspaces of L p (ℝn), 0 < p < 1

We prove that each translation and dilation invariant subspace X ⊂ L ^p (ℝⁿ), X ≠ L ^p (ℝⁿ), is contained in a maximal translation and dilation invariant subspace of L ^p (ℝⁿ). Moreover, we prove that the set of all maximal translation and dilation invariant subspaces of L ^p (ℝⁿ) has the power of continuum. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 345, 2007, pp. 5–21.     

Existence result for minimal hypersurfaces¶with a prescribed finite number of planar ends

Paralleling what has been done for minimal surfaces in ℝ³, we develop a gluing procedure to produce, for any k≥ 2 and any n≥ 3 complete immersed minimal hypersurfaces of ℝ ^{n +1} which have k planar ends. These surfaces are of the topological type of a sphere with k punctures and they all have finite total curvature. Received: 1 July 1999 / Revised version: 31 May 2000