FBI transforms in Gevrey classes

In this work we develop the FBI Transform tools in Gevrey classes. Our goal is to extend to a Gevrey-s obstacle withs < 3 the localization of poles result obtained by Sjöstrand [10] in the analytic class. In that work, the author proved that the pole-free zone is controlled by a constantC _0,a (which was only implicit in Bardos-Lebeau-Rauch [1]), improving the constantC _0,∞ of the results of Hargé-Lebeau [13] and Sjöstrand-Zworski [13] valid in C^∞ The works [3], [13] and [10] feature an adapted complex scaling for convex obstacles, but in [10] there is the addition of a small complex “G³ deformation”. The study of such Gevrey deformations for operators with symbols in Gevrey classes is the central point of this work.     

Cauchy problem for equations with multiple characteristics

This paper contains a proof of γ^n(χ) correctness of the noncharacteristic Cauchy problem for nonstrictly hyperbolic equations with analytic coefficients under the condition that its characteristic roots are smooth and under some additional assumptions on the lower-order terms. There are two extreme cases: (1) $\text{χ <}\text{r}\text{r}\text{? 1}$. In this case condition (0.6) is “void,” and we do not require conditions on P_s for s < m. For this case, see [3, 8]. (2) Case of constant multiplicity of characteristic roots and χ = +∞. In this case condition (0.6) implies conditions on P_s, where s = m, m ? 1,…, m ? r + 1, i.e., up to the same order as the necessary condition for C^∞-correctness [2]. Recall that in the case of equations with characteristics of constant multiplicity condition (0.6) (Levi's condition in this case) for χ = ∞ is necessary [2, 4] and sufficient [1] for C^∞-correctness.     

Courbure scalaire prescrite sur une variété C∞ compacte de dimension 2

Let (V₂, g) be a C^∞ compact Riemannian manifold of negative scalar curvature of dimension 2, and let f be a C^∞ function defined on V₂. We intend to find a condition on f in order that f be the scalar curvature of a metric conformal to the initial metric g.     

Dual spaces of C∞ (X)

ForX a locally compact Stonian Space, letC _∞ (X) denote the universally complete Riesz space of all extended-real-valued continuous functionsf onX for which {x∈X| |f (x)|=∞} is nowhere dense. In this paper the dual spaces ofC _∞ (X) (i.e. the spaces of order bounded; of σ-order continuous; of order continuous linear forms onC _∞ (X), and the extended order dual ofC _∞ (X) denote here byC _∞ (X)^ρ (introduced by W.A.J. Luxemburg and J.J. Masterson)) are characterized. It is shown thatC _∞ (X)^ρ can be identified in a canonical way with the inductive limitM _q (X) of the Riesz spaces of all normal Radon measures defined on the dense open subsets ofX. More generally, ifY is a locally compact space thenM _q (Y) is the extended order dual of the inductive limit of the Riesz spaces of all real-valued continuous functions defined on the dense open subsets ofY. IfX is locally compact and hyperstonian, then it is proved thatC _∞ (X) andC _∞ (X)^ρ are isomorphic, and a criterion forC _∞ (X)^ρ to be the universal completion of the space of order continuous linear forms onC _∞ (X) is given.     

On the closedness of the algebraic difference of closed convex sets

We characterize in a reflexive Banach space all the closed convex sets C₁ containing no lines for which the condition C₁^∞∩C₂^∞={0} ensures the closedness of the algebraic difference C₁−C₂ for all closed convex sets C₂. We also answer a closely related problem: determine all the pairs C₁, C₂ of closed convex sets containing no lines such that the algebraic difference of any sufficiently small uniform perturbations of C₁ and C₂ remains closed. As an application, we state the broadest setting for the strict separation theorem in a reflexive Banach space.     

Uniform approximation by some Hermite interpolating splines

In this paper some upper bound for the error ∥ s-f ∥_∞ is given, where f ε C¹[a,b], but s is a so-called Hermite spline interpolant (HSI) of degree 2q ?1 such that f(x_i) = s(x_i), f′(rmx_i) = s′(x_i), s^(j) (x_i) = 0 (i = 0, 1, …, n; j = 2, 3, …, q ?1; n > 0, q > 0) and the knots x_i are such that a = x₀ < x₁ < … < x_n = b. Necessary and sufficient conditions for the existence of convex HSI are given and upper error bound for approximation of the function fε C¹[a, b] by convex HSI is also given.     

Paley-Wiener theorems on groups of split rank one

Let G be a linear semisimple Lie group of split rank one with K a maximal compact subgroup. In this paper, we consider the space C_c^∞(G:F) of all functions in C_c^∞(G) whose left and right K-translates span a finite-dimensional space. Using the analytic continuation of the principal series to define the Fourier transform, we give a characterization of the Fourier transform of the space C_c^∞(G:F). This gives an analog of the classical Paley-Wiener theorem which gives a characterization of the Fourier transform of the space C_c^∞(Rⁿ).     

Fundamental sets of continuous functions on spheres

LetS ^m andS ^∞ denote the unit spheres inR m+1 ande ², respectively. We look for functionsf inC[?1, 1] such that the family of functionsx →f(<s,v>) is fundamental in the spaceC(S ^m). Herev runs overS ^m. There is a similar question forC(S ^∞), when this space is given the topology of uniform convergence on compact sets.     

Diffusions conditionnelles. I. Hypoellipticité partielle

This paper is devoted to the proof of certain results on conditional diffusions. By using the Malliavin calculus of variations, it is shown that, under minimal conditions on the vectors fields which define a two component diffusion (x_t, z_t) and their Lie brackets, a.s., for any T ? 0, the conditional law of x_t (t > 0), given $\text{B}$(z_s,s?T) a C^∞ density.     

Gevrey global solvability of non-singular real first-order differential operators

In this article we deal with Gevrey global solvability of non-singular first-order operators defined on an n-dimensional s-Gevrey manifold, s > 1. As done by Duistermaat and Hörmander in the C ^∞ framework, we show that Gevrey global solvability is equivalent the existence of a global cross section.     

Magic labelings of triangles

The first proof is given that for every even integer s≥4, the graph consisting of s vertex disjoint copies of C₃, (denoted sC₃) is vertex-magic. Hence it is also edge-magic. It is shown that for each even integer s≥6, sC₃ has vertex-magic total labelings with at least 2s−2 different magic constants. If s≡2mod4, two extra vertex-magic total labelings with the highest possible and lowest possible magic constants are given. If s=2⋅3^k, k≥1, it is shown that sC₃ has a vertex-magic total labeling with magic constant h if and only if (1/2)(15s+4)≤h≤(1/2)(21s+2). It is also shown that 2C₃ is not vertex-magic. If s is odd, vertex-magic total labelings for sC₃ with s+1 different magic constants are provided.     

The automorphisms of convolution algebras of C∞ functions

We find the automorphisms and the spectra of several different topological convolution algebras of C^∞-functions on the real line. Starting with the convolution algebra of compactly supported C^∞-functions, equipped with the usual LF-topology, we define a corresponding convolution algebra of C^∞-functions of arbitrarily fast exponential decay at ∞; and convolution algebras of a given finite degree r of exponential decay at ∞. These algebras may be described topologically as “hyper Schwartz spaces.” With a natural Frechet topology, which we define, they get a structure as locally m-convex algebras. The continuous automorphisms and spectra of these algebras are described completely. We show that the algebra of C^∞-functions of infinitly fast exponential decay at ∞, $\text{H J}$, on the one hand, and the algebra of C^∞-functions of only a finite degree $\text{e}{}^{\mathrm{?r¦x¦}}$ decay at ∞, $\text{J}$_r⁰, on the other hand, have quite different automorphisms, although $\text{H J}$ = ∩_r$\text{J}$_r⁰. As an application, we show that the conformal group is canonically represented as the full group of automorphisms of $\text{J}$_r⁰, and that this representation does not extend to a representation on the Banach algebra L¹($\text{R}$).     

The Atiyah bundle and connections on a principal bundle

Let M be a C ^∞ manifold and G a Lie a group. Let E _G be a C ^∞ principal G-bundle over M. There is a fiber bundle C(E _G ) over M whose smooth sections correspond to the connections on E _G . The pull back of E _G to C(E _G ) has a tautological connection. We investigate the curvature of this tautological connection.     

Some relations betweens-numbers of operators on Banach spaces

We considers-number sequences {s _n (T)} _n=1 ^∞ of linear and continuous operatorsT on Banach spaces and prove product formulas for the operator ideals ? _p,u ^(s) . Furthermore, we investigate the relationship between the eigenvalues of a Riesz operator and its Hilbert numbers.     

Positive periodic solutions for an integrodifferential model of mutualism

Sufficient conditions are obtained for the existence of positive periodic solutions of the following integrodifferential model of mutualism:

where r_i, K_i, α_i, σ_i, i = 1,2 are positive continuous ω-periodic functions, α_i > K_i, i = 1, 2, J_i ϵ C([0, ∞], [0, ∞)), and ∫^∞₀ J_i(s) ds = 1, i = 1, 2.     

Weak convergence of the past and future of Brownian motion given the present

In this paper, we show that for t > 0, the joint distribution of the past {W _t?s : 0 ≤ s ≤ t} and the future {W _{t + s} :s ≥ 0} of a d-dimensional standard Brownian motion (W _s ), conditioned on {W _t ∈ U}, where U is a bounded open set in ? ^d , converges weakly in C[0,∞)×C[0,∞) as t→∞. The limiting distribution is that of a pair of coupled processes Y + B ¹,Y + B ² where Y,B ¹,B ² are independent, Y is uniformly distributed on U and B ¹,B ² are standard d-dimensional Brownian motions. Let σ _t ,d _t be respectively, the last entrance time before time t into the set U and the first exit time after t from U. When the boundary of U is regular, we use the continuous mapping theorem to show that the limiting distribution as t → ∞ of the four dimensional vector with components \((W_{\sigma _{t}},t-\sigma _{t},W_{d_{t}},d_{t}-t)\), conditioned on {W _t ∈U}, is the same as that of the four dimensional vector whose components are the place and time of first exit from U of the processes Y + B ¹ and Y + B ² respectively.     

The isometries ofC p n,m intoC P

We characterize the isometries ofC _p ^n,m intoC _P (l≦p<∞, p ≠ 2, 2≦n,m). In particular, we find that ifX ?C _P is isometric toC _p ^n,m , then there exists a contractive projection from C_p ontoX.     

Perturbation and Interpolation Theorems for the H ∞-Calculus with Applications to Differential Operators

We prove comparison theorems for the H ^∞-calculus that allow to transfer the property of having a bounded H ^∞-calculus from one sectorial operator to another. The basic technical ingredient are suitable square function estimates. These comparison results provide a new approach to perturbation theorems for the H ^∞-calculus in a variety of situations suitable for applications. Our square function estimates also give rise to a new interpolation method, the Rademacher interpolation. We show that a bounded H ^∞-calculus is characterized by interpolation of the domains of fractional powers with respect to Rademacher interpolation. This leads to comparison and perturbation results for operators defined in interpolation scales such as the L _p -scale. We apply the results to give new proofs on the H ^∞-calculus for elliptic differential operators, including Schrödinger operators and perturbed boundary conditions. As new results we prove that elliptic boundary value problems with bounded uniformly coefficients have a bounded H ^∞-calculus in certain Sobolev spaces and that the Stokes operator on bounded domains Ω with ?Ω ∈ C ^1,1 has a bounded H ^∞-calculus in the Helmholtz scale L _p,σ (Ω), p ∈ (1,∞).     

Existence and uniqueness of C0-semigroup in L∞: a new topological approachExistence et unicité de C0-semigroupe sur L∞

A sub-Markov semigroup in L^∞ is in general not strongly continuous with respect to the norm topology. We introduce a new topology on L^∞ for which the usual sub-Markov semigroups in the literature become C₀-semigroups. This is realized by a natural extension of the Phillips theorem about dual semigroup. A simplified Hille–Yosida theorem is furnished. Moreover this new topological approach will allow us to introduce the notion of L^∞-uniqueness of pre-generator. We present several important pre-generators for which we can prove their L^∞-uniqueness. To cite this article: L. Wu, Y. Zhang, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 699–704.     

Noetherian varieties in definably complete structures

We prove that the zero-set of a C ^∞ function belonging to a noetherian differential ring M can be written as a finite union of C ^∞ manifolds which are definable by functions from the same ring. These manifolds can be taken to be connected under the additional assumption that every zero-dimensional regular zero-set of functions in M consists of finitely many points. These results hold not only for C ^∞ functions over the reals, but more generally for definable C ^∞ functions in a definably complete expansion of an ordered field. The class of definably complete expansions of ordered fields, whose basic properties are discussed in this paper, expands the class of real closed fields and includes o-minimal expansions of ordered fields. Finally, we provide examples of noetherian differential rings of C ^∞ functions over the reals, containing non-analytic functions.