Test elements for endomorphisms of free groups and algebras

LetE be a bounded Borel subset of ℝⁿ,n≥2, of positive Lebesgue measure andP _E the corresponding ‘Pompeiu transform”. We prove thatP _E is injective onL ^p(ℝⁿ) if 1≤p≤2n/(n-1). We explore the connection between this problem and a Wiener-Tauberian type theorem for theM(n) action onL ^q(ℝⁿ) for various values ofq. We also take up the question of whenP _E is injective in caseE is of finite, positive measure, but is not necessarily a bounded set. Finally, we briefly look at these questions in the contexts of symmetric spaces of compact and non-compact type.     

Riesz transformations of distributions and a generalized Hardy space

LeiE(ℝⁿ) be the space of all functions on ℝⁿ which can continue to the entire holomorphic functions on ℂⁿ. We define Riesz transformation R_j of distributions as a linear transformation of the quotient spaceD′(ℝⁿ)/E(ℝⁿ) to itself, j=1,2,..., n. These generalized Riesz transformations share the same properties with the classical ones, such as . As applications we generalize further a theorem of F. & M. Riesz generalized by Stein and Weiss, and then define a generalized Hardy space, of which some properties are studied.     

A homological characterization of hyperbolic groups

A finitely presented group G is hyperbolic iff H ⁽¹⁾ ₁(G,ℝ)=0=⁽¹⁾ ₂(G, ℝ), where H ⁽¹⁾ _* (resp. ⁽¹⁾ _*) denotes the ℓ₁-homology (resp. reduced ℓ₁-homology). If Γ is a graph, then every ℓ₁ 1-cycle in Γ with real coefficients can be approximated by 1-cycles of compact support. A 1-relator group G is hyperbolic iff H ⁽¹⁾ ₁(G,ℝ)=0. Oblatum: 30-IV-1997 & 14-V-1998 / Published online: 14 January 1999     

Uniform harmonic approximation with continuous extension to the boundary

Let Ω be an open set in ℝ ⁿ andE be a relatively closed subset of Ω. Further, letC _e(E) be the collection of real-valued continuous functions onE which extend continuously to the closure ofE in ℝ ⁿ . We characterize those pairs (Ω,E) which have the following property: every function inC _e(E) which is harmonic onE ⁰ can be uniformly approximated onE by functions which are harmonic on Ω and whose restrictions toE belong toC _e(E).     

On periodic solutions of even-order ordinary differential equations

Optimal in a certain sense sufficient conditions are given for the existence and uniqueness of ω-periodic solutions of the nonautonomous ordinary differential equation u ^(2m) =f(t,u,...,u ^(m-1) ), where the function f:ℝ×ℝ ^m →ℝ is periodic with respect to the first argument with period ω. Received: December 21, 1999; in final form: August 12, 2000?Published online: October 2, 2001     

A direct proof for lower semicontinuity of polyconvex functionals

Lower semicontinuity for polyconvex functionals of the form ∫_Ω g(detDu)dx with respect to sequences of functions fromW ^1,n (Ω;ℝ ⁿ ) which converge inL ¹ (Ωℝ ⁿ ) and are uniformly bounded inW ^1,n−1 (Ω;ℝ ⁿ ), is proved. This was first established in [5] using results from [1] on Cartesian Currents. We give a simple direct proof which does not involve currents. We also show how the method extends to prove natural, essentially optimal, generalizations of these results. Supported by MURST, Gruppo Nazionale 40% Partially supported by Australian Research Council     

Estimates of the periods of periodic points for nonexpansive operators

Suppose thatE is a finite-dimensional Banach space with a polyhedral norm ‖·‖, i.e., a norm such that the unit ball inE is a polyhedron. ℝ ⁿ with the sup norm or ℝ ⁿ with thel ₁-norm are important examples. IfD is a bounded set inE andT:D→D is a map such that ‖T(y)−T(z)‖≤ ‖y−z‖ for ally andz inE, thenT is called nonexpansive with respect to ‖·‖, and it is known that for eachx ∈D there is an integerp=p(x) such that lim _j→∞ T ^jp (x) exists. Furthermore, there exists an integerN, depending only on the dimension ofE and the polyhedral norm onE, such thatp(x)≤N: see [1,12,18,19] and the references to the literature there. In [15], Scheutzow has raised a question about the optimal choice ofN whenE=ℝ ⁿ ,D=K ⁿ , the set of nonnegative vectors in ℝ ⁿ , and the norm is thel ₁-norm. We provide here a reasonably sharp answer to Scheutzow’s question, and in fact we provide a systematic way to generate examples and use this approach to prove that our estimates are optimal forn≤24. See Theorem 2.1, Table 2.1 and the examples in Section 3. As we show in Corollary 2.3, these results also provide information about the caseD=ℝ ⁿ , i.e.,T:ℝ ⁿ →ℝ ⁿ isl ₁-nonexpansive. In addition, it is conjectured in [12] thatN=2 ⁿ whenE=ℝ ⁿ and the norm is the sup norm, and such a result is optimal, if true. Our theorems here show that a sharper result is true for an important subclass of nonexpansive mapsT:(ℝ ⁿ ,‖ · ‖_∞)→(ℝ ⁿ ,‖ · ‖_∞). Partially supported by NSF DMS89-03018.     

Automatic continuity of homomorphisms and fixed points on metric compacta

Let G₂(ℝ) × Sp₆(ℝ) and G₂(ℝ) × F₄(ℝ) be split dual pairs in split E₇(ℝ) and E₈(ℝ), respectively. It is known that the exceptional correspondences for these dual pairs are functorial on the level of infinitesimal characters. In this paper we show that these dual pair correspondences are functorial for the minimal K-types of principal series representations. Gordan Savin’s research is partially supported be NSF Grant no. DMS-0551846     

Enveloping Σ-C *-algebra of a smooth Frechet algebra crossed product by ℝ,K-theory and differential structure inC *-algebras

Given anm-tempered strongly continuous action α of ℝ by continuous^*-automorphisms of a Frechet^*-algebraA, it is shown that the enveloping ↡-C ^*-algebraE(S(ℝ, A^∞, α)) of the smooth Schwartz crossed productS(ℝ,A ^∞, α) of the Frechet algebra A^∞ of C^∞-elements ofA is isomorphic to the Σ-C ^*-crossed productC ^*(ℝ,E(A), α) of the enveloping Σ-C ^*-algebraE(A) ofA by the induced action. WhenA is a hermitianQ-algebra, one getsK-theory isomorphismRK _*(S(ℝ, A^∞, α)) =K ^*(C ^*(ℝ,E(A), α) for the representableK-theory of Frechet algebras. An application to the differential structure of aC ^*-algebra defined by densely defined differential seminorms is given.     

Algebraically constructible functions and signs of polynomials

LetW be a real algebraic set. We show that the following families of integer-valued functions onW coincide: (i) the functions of the formω →λ(X _ω ), where X _ω are the fibres of a regular morphismf :X →W of real algebraic sets, (ii) the functions of the formω →χ(X _ω ), where X _ω are the fibres of a proper regular morphismf :X →W of real algebraic sets, (iii) the finite sums of signs of polynomials onW. Such functions are called algebraically constructible onW. Using their characterization in terms of signs of polynomials we present new proofs of their basic functorial properties with respect to the link operator and specialization. Research partially supported by an Australian Research Council Small Grant. Second author also partially supported by KBN 610/P3/94.     

Decompositions of the Hardy space Hz^1（Ω）

We give a decomposition of the Hardy space Hz^1（Ω） into ＂div-curl＂ quantities for Lipschitz domains in R^n. We also prove a decomposition of Hz^1（Ω） into Jacobians det Du, u ∈ W0^1,2 （Ω,R^2） for Ω in R^2. This partially answers a well-known open problem.     

Existence principles for higher-order nonlocal boundary-value problems and their applications to singular Sturm-Liouville problems

We present existence principles for the nonlocal boundary-value problem (φ(u^(p−1)))′=g(t,u,...,u^(p−1), α_k(u)=0, 1≤k≤p−1, where p ≥ 2, π: ℝ → ℝ is an increasing and odd homeomorphism, g is a Carathéodory function that is either regular or has singularities in its space variables, and α _k: C ^p−1[0, T] → ℝ is a continuous functional. An application of the existence principles to singular Sturm-Liouville problems (−1)ⁿ(φ(u⁽²ⁿ⁻⁾))′=f(t,u,...,u⁽²ⁿ⁻¹⁾), u^(2k)(0)=0, α_ku^(2k)(T)+b_ku^(2k=1)(T)=0, 0≤k≤n−1, is given. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 240–259, February, 2008.     

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

Homoclinic Orbits of Nonperiodic Super Quadratic Hamiltonian System

This paper is concerned with solutions of the Hamiltonian system: [(u)\dot]=JH_u(t,u)\dot{u}=\mathcal{J}H_{u}(t,u) , where H(t,u)=\frac12u·Lu+W(t,u)H(t,u)=\frac{1}{2}u\cdot Lu+W(t,u) with L being a 2N×2N symmetric matrix and W∈C ¹(ℝ×ℝ^2N ,ℝ) being super quadratic at infinity in u∈ℝ^2N . We use variational methods to study this problem. By virtue of some auxiliary system related to the “limit equation” of the Hamiltonian system, we constructed linking levels of the variational functional Φ such that the minimax value E _l based on the linking structure of Φ satisfies 0 < E_l < E_l00E_l0E_{l_{0}} is the least action of the “limit equation”. Thus we can show the (PS) _c -condition holds true for all c < E_l0c相似文献   

Non-classical Godeaux surfaces

A non-classical Godeaux surface is a minimal surface of general type with χ = K ² = 1 but with h ⁰¹ ≠ 0. We prove that such surfaces fulfill h ⁰¹ = 1 and they can exist only over fields of positive characteristic at most 5. Like non-classical Enriques surfaces they fall into two classes: the singular and the supersingular ones. We give a complete classification in characteristic 5 and compute their Hodge-, Hodge–Witt- and crystalline cohomology (including torsion). Finally, we give an example of a supersingular Godeaux surface in characteristic 5.     

Properties of the Flows Generated by Stochastic Equations with Reflection

We consider the properties of a random set ϕ _t (ℝ ₊ ^d ), where ϕ _t (x) is a solution of a stochastic differential equation in ℝ ₊ ^d with normal reflection from the boundary that starts from a point x. We characterize inner and boundary points of the set ϕ _t (ℝ ₊ ^d ) and prove that the Hausdorff dimension of the boundary ∂ϕ _t (ℝ ₊ ^d ) does not exceed d − 1. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 8, pp. 1069 – 1078, August, 2005.     

Sums of squares on real algebraic surfaces

Consider real polynomials g₁, . . . , g_r in n variables, and assume that the subset K = {g₁≥0, . . . , g_r≥0} of ℝⁿ is compact. We show that a polynomial f has a representation in which the s_e are sums of squares, if and only if the same is true in every localization of the polynomial ring by a maximal ideal. We apply this result to provide large and concrete families of cases in which dim (K) = 2 and every polynomial f with f|_K≥0 has a representation (*). Before, it was not known whether a single such example exists. Further geometric and arithmetic applications are given. Support by DFG travel grant KON 1823/2002 and by the European RAAG network HPRN-CT-2001-00271 is gratefully acknowledged. Part of this work was done while the author enjoyed a stay at MSRI Berkeley. He would like to thank the institute for the invitation and the very pleasant working conditions.     

Incompleteness and minimality of complex exponential system

A necessary and sufficient condition is obtained for the incompleteness of a complex exponential system E(A,M)in C_α,where C_αis the weighted Banach space consisting of all complex continuous functions f on the real axis R with f(t)exp(-α(t))vanishing at infinity,in the uniform norm‖f‖_α=sup{|f(t)e~(-α(t))|:t∈R}with respect to the weightα(t).If the incompleteness holds, then the complex exponential system E(?)is minimal and each function in the closure of the linear span of complex exponential system E(?)can be extended to an entire function represented by a Taylor-Dirichlet series.     

Some remarks on the Hardy-Littlewood maximal function on variable <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> spaces

We show that any pointwise multiplier for BMO(ℝⁿ) generates a function p from the class (ℝⁿ) of those functions for which the Hardy-Littlewood maximal operator is bounded on the variable L^p space. In particular, this gives a positive answer to Diening's conjecture saying that there are discontinuous functions which nevertheless belong to (ℝⁿ).