A dyadic decomposition approach to a finitely degenerate hyperbolic problem

Abstract We use the Littlewood-Paley decomposition technique to obtain a C^∞-well-posedness result for a weakly hyperbolic equation with a finite order of degeneration. Keywords: Littlewood-Paley decomposition, Hyperbolic equations, C^∞-well-posedness, Approximate energy method     

On 𝒪ℒ∞ structures of nuclear C * -algebras

 We study the local operator space structure of nuclear C ^* -algebras. It is shown that a C ^* -algebra is nuclear if and only if it is an 𝒪ℒ_∞,λ space for some (and actually for every) λ>6. The 𝒪ℒ_∞ constant λ provides an interesting invariant

On the singularities of solutions of partial differential equations with constant coefficients

LetP be a differential operator with constant coefficients in ℝ ⁿ . Ifu is a distribution, the singular support ofu is the complement of the largest set whereu ∈C ^∞. Necessary and sufficient conditinos are obtained for a closed convex set Γ to be equal to the singular support ofu for someu withPu ∈C ^∞ or, equivalently, for Γ to contain the singular support ofu for someu withPu ∈C ^∞ butu ∉C ^∞. Related local uniqueness theorems analogous to the Holmgren theorem with supports replaced by singular supports are also given, as well as applications concerningP-convexity with respect to singular supports.     

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.     

Germes de d\'etermination infinie dans des classes lisses

We give sufficient conditions of infinite determinacy (in the sense of Mather) with respect to right equivalence, in subrings of C ^∞ (ℝ ⁿ ,0) defined by estimates on successive derivatives, such as rings of Gevrey germs. In this quantitative setting, a defect (hidden in the classical C ^∞ case) appears between the regularity of equivalent germs and the regularity of local diffeomorphisms of (ℝⁿ,0) giving the equivalence. Our conditions yield precise estimates of this defect, related to some suitable Łojasiewicz exponents of critical loci. We also show that the result is sharp in general. Received: Received: 27 May 1998     

Adjoining an Order Unit to a Matrix Ordered Space

We prove that an order unit can be adjoined to every L ^∞-matricially Riesz normed space. We introduce a notion of strong subspaces. The matrix order unit space obtained by adjoining an order unit to an L ^∞-matrically Riesz normed space is unique in the sense that the former is a strong L ^∞-matricially Riesz normed ideal of the later with codimension one. As an application of this result we extend Arveson’s extension theorem to L ^∞-matircially Riesz normed spaces. As another application of the above adjoining we generalize Wittstock’s decomposition of completely bounded maps into completely positive maps on C ^*-algebras to L ^∞-matricially Riesz normed spaces. We obtain sharper results in the case of approximate matrix order unit spaces. Mathematics Subject Classification (2000). Primary 46L07     

Microlocal Analysis of Generalized Pullbacks of?Colombeau Functions

In distribution theory the pullback of a general distribution by a C ^∞-function is well-defined whenever the normal bundle of the C ^∞-function does not intersect the wave front set of the distribution. However, the Colombeau theory of generalized functions allows for a pullback by an arbitrary c-bounded generalized function. It has been shown in previous work that in the case of multiplication of Colombeau functions (which is a special case of a C ^∞ pullback), the generalized wave front set of the product satisfies the same inclusion relation as in the distributional case, if the factors have their wave front sets in favorable position. We prove a microlocal inclusion relation for the generalized pullback (by a c-bounded generalized map) of Colombeau functions. The proof of this result relies on a stationary phase theorem for generalized phase functions, which is given in the Appendix. Furthermore we study an example (due to Hurd and Sattinger), where the pullback function stems from the generalized characteristic flow of a partial differential equation.      

C ∞-isometric imbeddings of the hyperbolic plane and of cylinders with hyperbolic metric in spherical spaces

Summary C^∞ — isometric imbeddings of the hyperbolic plane and of the two types of orientable cylinders with hyperbolic metric in spherical8-space are constructed, furthermore C^∞ — isometric imbeddings of the non-orientable cylinder with hyperbolic metric in Euclidean8-space and in spherical10-space. To Enrico Bompiani on his scientific Jubilee.     

Smooth roots of hyperbolic polynomials with definable coefficients

We prove that the roots of a definable C ^∞ curve of monic hyperbolic polynomials admit a definable C ^∞ parameterization, where ‘definable’ refers to any fixed o-minimal structure on (ℝ,+, ·). Moreover, we provide sufficient conditions, in terms of the differentiability of the coefficients and the order of contact of the roots, for the existence of C ^p (for p ∈ ℕ) arrangements of the roots in both the definable and the non-definable case. These conditions are sharp in the definable and, under an additional assumption, also in the non-definable case. In particular, we obtain a simple proof of Bronshtein’s theorem in the definable setting. We prove that the roots of definable C ^∞ curves of complex polynomials can be desingularized by means of local power substitutions t ↦ ±t ^N . For a definable continuous curve of complex polynomials we show that any continuous choice of roots is actually locally absolutely continuous.     

The affine complete hypersurfaces of constant Gauss-Kronecker curvature

Given a bounded convex domain Ω with C∞ boundary and a function ψ∈C∞（δΩ）, Li-Simon-Chen can construct an Euclidean complete and W-complete convex hypersurface M with constant affine Gauss-Kronecker curvature, and they guess the M is also affine complete. In this paper, we give a confirmation answer.     

Approximation of (ψ, β)-differentiable functions by Poisson integrals in the uniform metric

We obtain asymptotic equalities for upper bounds of approximations of functions from the class C _β,∞^ψ by Poisson integrals in the metric of the space C.     

Non-oscillatory criteria for a class of second order non-linear forced neutral-delay differential equations

In this paper, sufficient conditions are obtained, so that the second order neutral delay differential equation

Integration of differential forms on manifolds with locally-finite variations

It is well known that one can integrate any compactly supported, continuous, differential n-form over n-dimensional C¹-manifolds in ℝ^m (m ≥ n). For n = 1, the integral may be defined over any locally rectifiable curve. Another generalization is the theory of currents (linear functionals on the space of compactly supported C^∞-differential forms). The theme of the article is integration of measurable differential n-forms over n-dimensional n C⁰-manifolds in ℝ^m with locally-finite n-dimensional variations (a generalization of locally rectifiable curves to dimension n > 1). The main result states that any such manifold generates an n-dimensional current in ℝ^m (i.e., any compactly supported C^∞ n-form can be integrated over a manifold with the properties mentioned above). Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 327, 2005, pp. 168–206.     

Total Curvature for <Emphasis Type="Italic">C</Emphasis><Superscript>∞</Superscript>-Singular Surfaces with Limiting Tangent Bundle

The total curvature of a compact C^∞-immersed surface in Euclidean 3-space ³ can be interpreted as the average number of critical points for a linear ‘height’ function. The Morse inequalities provide an intrinsic topological lower bound for the total curvature and ‘tight’ surfaces, which realize equality, have been an active topic of research. The objective of this paper is to describe the natural notion of total curvature for C^∞-singular surfaces which fail to immerse on C^∞-embedded closed curves, but which have a C^∞-globally defined unit normal (e.g. caustics, or critical images for mappings of 3-manifolds into Euclidean 3-space). For such surfaces total curvature consists of a sum of two-dimensional and one-dimensional integrals, which have various lower bounds. Large sets of LT-surfaces which realize equality are then constructed. As an application, the orthogonal projection of an immersed tight hypersurface in Euclidean 4-space is shown to have LT-tight critical image, and several related inequalities are given. Mathematics Subject Classifications (2000): 57N65, 14P99, 53C21, 53B25, 53B20.     

Nonuniqueness for a bounded, differentiable complete orthonormal system inL 2[0, 1]

The argument of Müsegian and Ovsepjan is adapted to produce a complete orthonormal system on [0, 1] of uniformly bounded functions, differentiable on [0, 1], andC ^∞ on [0, 1], for which the analogue of Cantor's uniqueness theorem is false. We also construct a complete orthonormal system ofC ^∞ functions which vanish to infinite order at both endpoints.     

Cesàro-type operators on Hardy, BMOA and bloch spaces

Let H ^p, p ∈ (0, ∞], BMOA and B ^a, a ∈ (0, ∞) be the classical p-Hardy, analytic BMO(∂) (bounded mean oscillation on the unit circle) and a-Bloch space on the unit disk. In this paper, we prove that the Cesàro-type operator: C ^α, α ∈ (−1, ∞) is bounded on H ^p, p ∈ (0, ∞) and on B ^a, a ∈ (1, ∞), but, unbounded on H ^∞, BMOA and B ^a, a ∈ (0, 1]. In particular, we give an answer to the Stempak’s open problem.     

A Uniqueness Result for a Model for Mixtures in the Absence of External Forces and Interaction Momentum

We consider a continuum model describing steady flows of a miscible mixture of two fluids. The densities ϱ _i of the fluids and their velocity fields u ⁽ⁱ⁾ are prescribed at infinity: ϱ _i |_∞ = ϱ _i∞ > 0, u ⁽ⁱ⁾|∞ = 0. Neglecting the convective terms, we have proved earlier that weak solutions to such a reduced system exist. Here we establish a uniqueness type result: in the absence of the external forces and interaction terms, there is only one such solution, namely ϱ _i ≡ ϱ _i∞, u ⁽ⁱ⁾ ≡ 0, i = 1, 2. This work was supported by the SFB 611 at the University of Bonn and the European HYKE network (contract no. HPRN-CT-2002-00282). The third author was also supported by the project CSF 201/03/0934, and by MSM 0021620839.     

Polynomial hulls and an optimization problem

We say that a subset of Cⁿ is hypoconvex if its complement is the union of complex hyperplanes. We say it is strictly hypoconvex if it is smoothly bounded hypoconvex and at every point of the boundary the real Hessian of its defining function is positive definite on the complex tangent space at that point. Let B_n be the open unit ball in Cⁿ.Suppose K is a C^∞ compact manifold in ∂B₁ × Cⁿ, n > 1, diffeomorphic to ∂B₁ × ∂B_n, each of whose fibers K_z over ∂B₁ bounds a strictly hypoconvex connected open set. Let K be the polynomialhull of K. Then we show that K∖K is the union of graphs of analytic vector valued functions on B₁. This result shows that an unnatural assumption regarding the deformability of K in an earlier version of this result is unnecessary. Next, we study an H^∞ optimization problem. If pis a C^∞ real-valued function on ∂B₁× Cⁿ, we show that the infimum γ_ρ = inf_ƒ∈H ^∞ (B₁)ⁿ ‖ρ(z, ƒ (z))‖_∞ is attained by a unique bounded ƒ provided that the set (z, w) ∈ ∂B₁ × C ⁿ|ρ(z, w) ≤ γ_ρ has bounded connected strictly hypoconvex fibers over the circle.     

Oscillatory and asymptotic behaviour of a nonlinear second order neutral differential equation

In this paper, necessary and sufficient conditions for the oscillation and asymptotic behaviour of solutions of the second order neutral delay differential equation (NDDE)

On some special measures on βG

LetG be a discrete abelian group,Ĝ the character group ofG, andl ^∞(G)^* the conjugate ofl ^∞(G) equipped with an Arens product. In many cases, we can find unitary functionsf such that χf is almost convergent to zero for all χ∈Ĝ. Some of these functions are then used to produce elements μ∈l ^∞(G)^* such that γμ=0 whenever γ is an annihilator ofC ₀(G). Regarded as Borel measures on βG, these elements satisfyxμ=0 for allx∈βG/G. They belong to the radical ofl ^∞(G)^*, and each of them generates a left ideal ofl ^∞(G)^* that contains no minimal left ideal.