METRIC ENTROPY OF HOMEOMORPHISM ON NON-COMPACT METRIC SPACE

Let T : X → X be a uniformly continuous homeomorphism on a non-compact metric space (X, d). Denote by X* = X ∪ {x*} the one point compactification of X and T * : X* → X* the homeomorphism on X* satisfying T *|X = T and T *x* = x*. We show that their topological entropies satisfy hd(T, X) ≥ h(T *, X*) if X is locally compact. We also give a note on Katok’s measure theoretic entropy on a compact metric space.     

Orthogonal polynomials and cubature formulae on balls,simplices, and spheres

We report on recent developments on orthogonal polynomials and cubature formulae on the unit ball B^d, the standard simplex T^d, and the unit sphere S^d. The main result shows that orthogonal structures and cubature formulae for these three regions are closely related. This provides a way to study the structure of orthogonal polynomials; for example, it allows us to use the theory of h-harmonics to study orthogonal polynomials on B^d and on T^d. It also provides a way to construct new cubature formulae on these regions.     

Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations

We study nonlinear Schrödinger equations, posed on a three dimensional Riemannian manifold M. We prove global existence of strong H¹ solutions on M=S³ and M=S²×S¹ as far as the nonlinearity is defocusing and sub-quintic and thus we extend results of Ginibre, Velo and Bourgain who treated the cases of the Euclidean space R³ and the torus T³=R³/Z³ respectively. The main ingredient in our argument is a new set of multilinear estimates for spherical harmonics.     

Lifespan Theorem for simple constrained surface diffusion flows

We consider closed immersed hypersurfaces in R³ and R⁴ evolving by a special class of constrained surface diffusion flows. This class of constrained flows includes the classical surface diffusion flow. In this paper we present a Lifespan Theorem for these flows, which gives a positive lower bound on the time for which a smooth solution exists, and a small upper bound on the total curvature during this time. The hypothesis of the theorem is that the surface is not already singular in terms of concentration of curvature. This turns out to be a deep property of the initial manifold, as the lower bound on maximal time obtained depends precisely upon the concentration of curvature of the initial manifold in L² for M² immersed in R³ and additionally on the concentration in L³ for M³ immersed in R⁴. This is stronger than a previous result on a different class of constrained surface diffusion flows, as here we obtain an improved lower bound on maximal time, a better estimate during this period, and eliminate any assumption on the area of the evolving hypersurface.     

Derivations of non-separable C1-algebras

The question of which C¹-algebras have only inner derivations has been considered by a number of authors for 25 years. The separable case is completely solved, so this paper deals only with the non-separable case. In particular, we show that the C¹-tensor product of a von Neumann algebra and an abelian C¹-algebra has only inner derivations. Other special types of C¹-algebras are shown to have only inner derivations as well such as the C¹-tensor product of L(H) (all bounded operators on separable Hilbert space) and any separable C¹-algebra having only inner derivations. Derivations from a smaller C¹-algebra into a larger one are also considered, and this concept is generalized to include derivations between C¹-algebras connected by a ¹-homomorphism. Finally, we consider the general problem of a sequence of linear functionals on a C¹-algebra which converges to zero (in norm) when restricted to any abelian C¹-subalgebra. Does such a sequence converge to zero in norm? The answer is “yes” for normal functionals on L(H), but unknown in general.     

On the best constants in Markov-type inequalities involving Laguerre norms with different weights

The paper concerns best constants in Markov-type inequalities between the norm of a higher derivative of a polynomial and the norm of the polynomial itself. The norm of the polynomial is taken in L ² on the half-line with the weight t ^α e ^?t and the derivative is measured in L ² on the half-line with the weight t ^β e ^?t . Under an additional assumption on the difference β ? α, we determine the leading term of the asymptotics of the constants as the degree of the polynomial goes to infinity.     

Extreme points, support points and the Loewner variation in several complex variables

In this paper we consider extreme points and support points for compact subclasses of normalized biholomorphic mappings of the Euclidean unit ball Bn in Cn. We consider the class S0(Bn) of biholomorphic mappings on Bn which have parametric representation, i.e., they are the initial elements f (·, 0) of a Loewner chain f (z, t) = etz + ··· such that {e-tf (·, t)}t 0 is a normal family on Bn. We show that if f (·, 0) is an extreme point (respectively a support point) of S0(Bn), then e-tf (·, t) is an extreme point of S0(Bn) for t 0 (respectively a support point of S0(Bn) for t ∈[0, t0] and some t0 > 0). This is a generalization to the n-dimensional case of work due to Pell. Also, we prove analogous results for mappings which belong to S0(Bn) and which are bounded in the norm by a fixed constant. We relate the study of this class to reachable sets in control theory generalizing work of Roth. Finally we consider extreme points and support points for biholomorphic mappings of Bn generated by using extension operators that preserve Loewner chains.     

Inverse Scattering for a Schroedinger Operator with a Repulsive Potential

We consider a pair of Hamiltonians （H, H0） on L2（R^n）, where H0=p^2 -x^2 is a SchrSdinger operator with a repulsive potential, and H = H0＋V（x）. We show that, under suitable assumptions on the decay of the electric potential, V is uniquely determined by the high energy limit of the scattering operator.     

On interpolation of interpolating sequences

Let A be a uniform algebra on the compact space X and σ a probability measure on X. We define the Hardy spaces H^P(σ) and the H^P(σ) interpolating sequences S in the p-spectrum M_p of σ. Under some structural hypotheses on (A, σ), we prove that if a sequence S ⊂ M_p is H^P(σ) interpolating, then it is H^s(σ) interpolating for s < p. In the special case of the unit ball B of ?ⁿ this answers a natural question asked in [8].     

Multipliers of Sobolev spaces

Let W^m,p denote the Sobolev space of functions on $\text{R}$ⁿ whose distributional derivatives of order up to m lie in L^p($\text{R}$ⁿ) for 1 ? p ? ∞. When 1 < p < ∞, it is known that the multipliers on W^m,p are the same as those on L^p. This result is true for p = 1 only if n = 1. For, we prove that the integrable distributions of order ?1 whose first order derivatives are also integrable of order ?1, belong to the class of multipliers on W^m,1 and there are such distributions which are not bounded measures. These distributions are also multipliers on L^p, for 1 < p < ∞. Moreover, they form exactly the multiplier space of a certain Segal algebra. We have also proved that the multipliers on W^m,l are necessarily integrable distributions of order ?1 or ?2 accordingly as m is odd or even. We have obtained the multipliers from L¹($\text{R}$ⁿ) into W^m,p, 1 ? p ? ∞, and the multiplier space of W^m,1 is realised as a dual space of certain continuous functions on $\text{R}$ⁿ which vanish at infinity.     

On a duality for crossed products of C1-algebras

Let $\text{U}$ be a C¹-algebra, and G be a locally compact abelian group. Suppose α is a continuous action of G on $\text{U}$. Then there exists a continuous action $\text{}\text{?}$ga of the dual group $\text{G}\text{?}$ of G on the C¹-crossed product by α such that the C¹-crossed product is isomorphic to the tensor product and the C¹-algebra of all compact operators on L²(G).     

Uniqueness of analytic continuation: Necessary and sufficient conditions

Necessary and sufficient conditions for uniqueness of analytic continuation are investigated for a system of m ? 1 first-order linear homogeneous partial differential equations in one unknown, with complex-valued $\text{b}$^∞ coefficients, in some connected open subset of $\text{R}$^k, k ? 2. The type of system considered is one for which there exists a real k-dimensional, $\text{b}$^∞, connected C-R submanifold M^k of $\text{C}$ⁿ, for k, n ? 2, such that the system may be identified with the induced Cauchy-Riemann operators on M^k. The question of uniqueness of analytic continuation for a system of partial differential equations is thus transformed to the question of uniqueness of analytic continuation for C-R functions on the manifold M^k ? Cⁿ. Under the assumption that the Levi algebra of M^k has constant dimension, it is shown that if the excess dimension of this algebra is maximal at every point, then M^k has the property of uniqueness of analytic continuation for its C-R functions. Conversely, under certain mild conditions, it is shown that if M^k has the property of uniqueness of analytic continuation for all $\text{b}$^∞ C-R functions, and if the Levi algebra has constant dimension on all of M^k, then the excess dimension must be maximal at every point of M^k.     

On the Γ-Convergence of integral functionals defined on sobolev weakly connected spaces

We introduce and study the concept of Γ-convergence of functionateI _s :W ^k,m (Ω)→?,s=1,2,..., to a functional defined on (W ^k,m (Ω))² and describe the relationship between this type of convergence and the convergence of solutions of Neumann variational problems. For a sequence of integral functionateI _s :W ^k,m (Ω)→?, we prove a theorem on the selection of a subsequence Γ-convergent to an integral functional defined on (W ^k,m (Ω))².     

Completely inverse AG ∗∗-groupoids

A completely inverse AG ^??-groupoid is a groupoid satisfying the identities (xy)z=(zy)x, x(yz)=y(xz) and xx ^?1=x ^?1 x, where x ^?1 is a unique inverse of x, that is, x=(xx ^?1)x and x ^?1=(x ^?1 x)x ^?1. First we study some fundamental properties of such groupoids. Then we determine certain fundamental congruences on a completely inverse AG ^??-groupoid; namely: the maximum idempotent-separating congruence, the least AG-group congruence and the least E-unitary congruence. Finally, we investigate the complete lattice of congruences of a completely inverse AG ^??-groupoids. In particular, we describe congruences on completely inverse AG ^??-groupoids by their kernel and trace.     

Irreducible 1-representations of some group rings and associated Banach 1-algebras

Let G be the group which is a wreath product of two infinite cyclic groups. We construct a faithful, topologically irreducible 1-representation of l¹(G) on l²($\text{Z}$), and a faithful, strictly irreducible representation on l¹($\text{Z}$). The finite-dimensional, irreducible 1-representations of l¹(G) are described, and we find that an associated C¹-algebra is primitive and antiliminal, with a separating set of finite-dimensional, irreducible representations. A similar study is made of the group H generated by x and y with the relation x^?1yx = x².     

Lévy white noise measures on infinite-dimensional spaces: Existence and characterization of the measurable support

It is shown that a Lévy white noise measure Λ always exists as a Borel measure on the dual K^′ of the space K of C^∞ functions on R with compact support. Then a characterization theorem that ensures that the measurable support of Λ is contained in S^′ is proved. In the course of the proofs, a representation of the Lévy process as a function on K^′ is obtained and stochastic Lévy integrals are studied.     

Sur le changement de base stable des intégrales orbitales pondérées

Let G^′ be a quasi-split connected reductive group over a p-adic field F. Let E be a cyclic extension of F. In the context of cyclic base change, we can attach to G^′ and E a twisted space G^* (in the sense of Labesse). Let G be an inner form of G^*. If G^′ is GL(n), SL(n) or more generally a group which we call L-stable, we define and prove the existence of a non-invariant transfer between the weighted orbital integrals of G and those of G^′. For GL(n), such a transfer has been conjectured by Labesse. The proof is based on previous results of harmonic analysis on Lie algebras and on a generalization of a result of Waldspurger concerning Arthur's (G,M)-families.     

Pointwise Lipschitz functions on metric spaces

For a metric space X, we study the space D^∞(X) of bounded functions on X whose pointwise Lipschitz constant is uniformly bounded. D^∞(X) is compared with the space LIP^∞(X) of bounded Lipschitz functions on X, in terms of different properties regarding the geometry of X. We also obtain a Banach-Stone theorem in this context. In the case of a metric measure space, we also compare D^∞(X) with the Newtonian-Sobolev space N1,∞(X). In particular, if X supports a doubling measure and satisfies a local Poincaré inequality, we obtain that D^∞(X)=N1,∞(X).     

Littlewood–Paley and Spectral Multipliers on Weighted L p Spaces

Let L be a non-negative self-adjoint operator acting on L ²(X), where X is a space of homogeneous type. Assume that L generates a holomorphic semigroup e ^?tL whose kernel p _t (x,y) has a Gaussian upper bound but there is no assumption on the regularity in variables x and y. In this article we study weighted L ^p -norm inequalities for spectral multipliers of L. We show that a weighted Hörmander-type spectral multiplier theorem follows from weighted L ^p -norm inequalities for the Lusin and Littlewood–Paley functions, Gaussian heat kernel bounds, and appropriate L ² estimates of the kernels of the spectral multipliers.