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     

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.     

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

Éclatement des coefficients des séries entières et deux théorèmes de Gabrielov

Given a non trivial power series in ℝ ^m × ℝ ^k , it is in general not possible to choose a good direction in ℝ ^k in order to apply Weierstrass Preparation Theorem. Now, one can make it possible by blowing-up coefficients in ℝ ^m . This enables e. g. to prove in some natural way Gabrielov’s complement theorem, as well as Gabrielov’s fiber components theorem in subanalytic geometry.      

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 a question of A. E. Nussbaum on measurability of families of closed linear operators in a Hilbert space

The purpose of this note is to answer a question A. E. Nussbaum formulated in 1964 about the possible equivalence between weak measurability of a family of densely defined, closed operators {T(t)} _t∈ℝ in a separable complex Hilbert space H\mathcal{H} on one hand, and the notion of measurability of the 2 × 2 operator-valued matrix of projections {(P(Γ(T(t))) _j,k )_1⩽j,k⩽2} _t∈ℝ onto the graph Γ(T(t)) of T(t) on the other, in the negative.     

A degenerate Hartman theorem

Consider a real analytic diffeomorphism,f:ℝ²→ℝ², withq as a non-hyperbolic fixed point andDf(q)=Id. Placing sufficient conditions on lowest-order non-linear terms in the expansion off, we show the function is topologically conjugate with a decoupled product map. The impetus for studying such a function arose in the classical three-body problem.     

On the total curvatures of a tame function

Given a definable function f: ℝ ⁿ ↦ ℝ, enough differentiable, we study the continuity of the total curvature function t → K(t), total curvature of the level f ⁻¹(t), and the total absolute curvature function t → |K|(t), total absolute curvature of the level f ⁻¹(t). We show they admits at most finitely many discontinuities. Partially supported by the European research network IHP-RAAG contract number HPRN-CT-2001-00271 and partially supported by Deutsche Forschungs-Gemeinschaft in the Priority Program Global Differential Geometry.     

Bernstein’s analyticity theorem for quantum differences

We consider real valued functions f defined on a subinterval I of the positive real axis and prove that if all of f’s quantum differences are nonnegative then f has a power series representation on I. Further, if the quantum differences have fixed sign on I then f is analytic on I.     

Affinely equivalent complete flat manifolds

Let E _Aff(Γ,G, m) be the set of affine equivalence classes of m-dimensional complete flat manifolds with a fixed fundamental group Γ and a fixed holonomy group G. Let n be the dimension of a closed flat manifold whose fundamental group is isomorphic to Γ. We describe E _Aff(Γ,G, m) in terms of equivalence classes of pairs (ε, ρ), consisting of epimorphisms of Γ onto G and representations of G in ℝ ^m-n . As an application we give some estimates of card E _Aff(Γ,G, m).     

Mutual isomorphisms of translations of a homogeneous flow

Ergodic one-parameter flows (G/Γ,g _ℝ) induced by the left action of a subgroupg _ℝ ⊂G on homogeneous spaces of finite volume are considered. Letℳ ⊂ ℝ⁺ be the set of allt>0 such that the cascade (G/Γ,g _t ℤ) is metrically isomorphic to the cascade (G/Γ,g _ℤ). We prove that eitherℳ is at most countable or the subgroupg _ℝ is horocyclic andℳ=ℝ⁺. We prove that a metric isomorphism of ergodic quasi-unipotent cascades (or flows) is affine on almost all fibers of a certain natural bundle. The result generalizes Witte's theorem on the affinity of such isomorphisms of cascades with the mixing property; this is applied to the study of the structure of the setℳ ⊂ ℝ⁺. The proof is based on the fundamental Ratner theorem stating that the ergodic measures of unipotent cascades are algebraic. Translated fromMatematicheskie Zametki, Vol. 58, No. 1, pp. 98–110, July, 1995. This research was partially supported by the Russian Foundation for Basic Research under grant No. 94-01-00082a and by the International Science Foundation.     

Relative version of the Titchmarsh convolution theorem

We consider the algebra C _u = C _u (ℝ) of uniformly continuous bounded complex functions on the real line ℝ with pointwise operations and sup-norm. Let I be a closed ideal in C _u invariant with respect to translations, and let ah _I (f) denote the minimal real number (if it exists) satisfying the following condition. If λ > ah _I (f), then ( [^(f)] - [^(g)] ) |_V = 0\left. {\left( {\hat f - \hat g} \right)} \right|_V = 0 for some g ∈ I, where V is a neighborhood of the point λ. The classical Titchmarsh convolution theorem is equivalent to the equality ah _I (f ₁ · f ₂) = ah _I (f ₁) + ah _I (f ₂), where I = {0}. We show that, for ideals I of general form, this equality does not generally hold, but ah _I (f ⁿ ) = n · ah _I (f) holds for any I. We present many nontrivial ideals for which the general form of the Titchmarsh theorem is true.     

On Taylor-like integral representations of analytic functions

Let f be an analytic function in a complex domain D and (without loss of generality) assume 0∈D. Then the paper’s aim is to derive a Taylor-like integral expression for f, i.e. an integral representation analogous to the corresponding power series, say, ∑ _k=0^∞ a _k t ^k /k!. We start from the simplest case f(t)=e ^t , which leads to the identity

Dirichlet forms for diffusion in ℝ<Superscript>2</Superscript> and jumps on fractals: The regularity problem

We define a Dirichlet form ɛ describing diffusion in ℝ ^d and jumps in a fractal Γ ⊂ ℝ ^d . The jump measure J is defined as an image of a jump measure j of a process in a non-Archimedean metric space. As the result the jump intensity depends on the hierarchical structure of Γ rather than the geometric distance in ℝ ^d . For a class of fractals in ℝ² we find a condition on the measure j so that the Dirichlet form ɛ is regular. The condition is given in terms of Hausdorff dimension of Γ.     

Logarithm laws for flows on homogeneous spaces

In this paper we generalize and sharpen D. Sullivan’s logarithm law for geodesics by specifying conditions on a sequence of subsets {A _t  | t∈ℕ} of a homogeneous space G/Γ (G a semisimple Lie group, Γ an irreducible lattice) and a sequence of elements f _t of G under which #{t∈ℕ | f _t x∈A _t } is infinite for a.e. x∈G/Γ. The main tool is exponential decay of correlation coefficients of smooth functions on G/Γ. Besides the general (higher rank) version of Sullivan’s result, as a consequence we obtain a new proof of the classical Khinchin-Groshev theorem on simultaneous Diophantine approximation, and settle a conjecture recently made by M. Skriganov. Oblatum 27-VII-1998 & 2-IV-1999 / Published online: 5 August 1999     

Cusp forms

LetG andH ⊂G be two real semisimple groups defined overQ. Assume thatH is the group of points fixed by an involution ofG. Letπ ⊂L ²(H\G) be an irreducible representation ofG and letf επ be aK-finite function. Let Γ be an arithmetic subgroup ofG. The Poincaré seriesP _f(g)=Σ_H∩ΓΓ f(γ{}itg) is an automorphic form on Γ\G. We show thatP _f is cuspidal in some cases, whenH ∩Γ\H is compact. Partially supported by NSF Grant # DMS 9103608.     

Existence, uniqueness and stability of C m solutions of iterative functional equations

In this paper we discuss a relatively general kind of iterative functional equation G(x,f(x), ...,f ⁿ (x)) = 0 (for allx ∈J), whereJ is a connected closed subset of the real number axis ℝ,G∈C ^m (J ⁿ⁺¹, ℝ) andn ≥ 2. Using the method of approximating fixed points by small shift of maps, choosing suitable metrics on functional spaces and finding a relation between uniqueness and stability of fixed points of maps of general spaces, we prove the existence, uniqueness and stability ofCm solutions of the above equation for any integer m ≥ 0 under relatively weak conditions, and generalize related results in reference in different aspects.     

On boundary behavior of generalized quasi-isometries

We establish a series of criteria for continuous and homeomorphic extension to the boundary of the so-called lower Q-homeomorphisms f between domains in [`(ⁿ)] = ⁿ è{ ￥} ,n 3 2\overline {{^n}} = {^{^n}} \cup \{ \infty \} ,n \ge 2, under integral constraints of the type ∫ Φ(Q ⁿ⁻¹(x))dm(x) < ∞ with a convex non-decreasing function Φ: [0,∞]→[0,∞]. Integral conditions on Φ are found that are necessary and sufficient for a continuous extension of f to the boundary. Our results are applied to finitely bi-Lipschitz mappings, which are a far-reaching generalization of isometries as well as quasi-isometries in ℝ ⁿ . In particular, a generalization and strengthening of the well-known theorem of Gehring-Martio on homeomorphic extension to boundaries of quasi-conformal mappings between QED (quasi-extremal distance) domains is obtained.