Connected components in the space of composition operators on<Emphasis Type="Italic">H∞</Emphasis> functions of many variables

LetE be a complex Banach space with open unit ballB _e. The structure of the space of composition operators on the Banach algebra H∞, of bounded analytic functions onB _e with the uniform topology, is studied. We prove that the composition operators arising from mappings whose range lies strictly insideB _e form a path connected component. WhenE is a Hilbert space or aC _o(X)- space, the path connected components are shown to be the open balls of radius 2. The research of this author was supported by grant number SAB1999-0214 from the Ministerio de Educación, Cultura y Deporte during his stay at the Universidad de Valencia. The research of this author was partially supported DGES(Spain) pr. 96-0758. The research of this author was partially supported by Magnus Ehrnrooths stiftelse.     

A generalized inductive limit strict topology on the space of bounded continuous functions

A generalized inductive limit strict topology β_∞ is defined on C_b(X, E), the space of all bounded, continuous functions from a zero-dimensional Hausdorff space X into a locally -convex space E, where is a field with a nontrivial and nonarchimedean valuation, for which is a complete ultrametric space. Many properties of the topology β_∞ are proved and the dual of (C_b (X, E), β_∞) is studied.     

Analytic functions of bounded mean oscillation and the Bloch space

In this paper we show that the closure of the space BMOA of analytic functions of bounded mean oscillation in the Bloch spaceB is the image P(U) of space of all continuous functions on the maximal ideal space ofH under the Bergman projection P. It is proved that the radial growth of functions in P(U) is slower than the iterated logarithm studied by Makarov. So some geometric conditions are given for functions in P(U), which we can easily use to construct many Bloch functions not in P(U).     

The structure of spaces of quasianalytic functions of Roumieu type

It is shown that spaces of quasianalytic ultradifferentiable functions of Roumieu type ℰ_{w}(Ω), on an open convex set , satisfy some new (Ω) -type linear topological invariants. Some consequences for the splitting of short exact sequences of these spaces as well as for the structure of the spaces are derived. In particular, Fréchet quotients of ℰ_{w}(Ω) have property (), while dual Fréchet quotients have property () of Vogt. The work of P. Domański was supported by Committee of Scientific Research (KBN), Poland, grant P03A 022 25.     

The transmission problem in domains with a corner point for the Laplace operator in weighted Hölder spaces

We prove the existence and uniqueness of a solution to the elliptic transmission problem in nonsmooth domains in the weighted Hölder space. The coercive estimates of the solution are given.     

Fundamental solution for the Q-Laplacian and sharp Moser-Trudinger inequality in Carnot groups

For a general Carnot group G with homogeneous dimension Q we prove the existence of a fundamental solution of the Q-Laplacian u_Q and a constant a_Q>0 such that exp(−a_Qu_Q) is a homogeneous norm on G. This implies a representation formula for smooth functions on G which is used to prove the sharp Carnot group version of the celebrated Moser-Trudinger inequality.     

Generalization of the Newman-Shapiro isometry theorem and Toeplitz operators

We give a generalization of the Newman-Shapiro Isometry Theorem to the case of Hilbert space-valued entire functions, which are square-summable with respect to the Gaussian measure on ⁿ , together with some applications in the theory of Toeplitz operators with operator-valued symbols. The study of various properties (such as density of domains, cores, closedness and boundedness from below) of these operators in illustrated with many relevant examples.Research supported by KBN under grant no. 2 P03A 041 10.     

Weighted L<Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-inequalities for multi-parameter Riesz type potentials and strong fractional maximal operators

We prove weighted L ^p -inequalities for multi-parameter Riesz type potentials, strong fractional maximal operators and their dyadic counterparts. Our proofs avoid the Good-λ inequalities used earlier in the R ^m -case and are based on our integrated multi-parameter summation by parts lemma, that might be of independent interest.     

Elliptic problems with a Hardy potential and critical growth in the gradient: Non-resonance and blow-up results

In this article we analyze existence and nonexistence of positive solutions to problem

     

Herz Classes and Toeplitz Operators in the Disk

In this paper we decompose into diadic annuli and consider the class S_p,q of Toeplitz operators T_φ for which the sequence of Schatten norms belongs to ℓ^q, where φ_n = φχ A_n. We study the boundedness and compactness of the operators in S_p,q and we describe the operators T_φ , φ ≥ 0 in these spaces in terms of weighted Herz norms of the averaging operator of the symbols φ.     

Elliptic operators with unbounded diffusion coefficients in L 2 spaces with respect to invariant measures

We study the self-adjoint and dissipative realization A of a second order elliptic differential operator with unbounded regular coefficients in , where μ(dx) = ρ (x)dx is the associated invariant measure. We prove a maximal regularity result under suitable assumptions, that generalize the well known conditions in the case of constant diffusion part. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

On the definition of a probabilistic normed space

Summary In this paper we give a new definition of a probabilistic normed space. This definition, which is based on a characterization of normed spaces by means of a betweenness relation, includes the earlier definition of A. N. erstnev as a special case and leads naturally to the definition of the principal class of probabilistic normed spaces, the Menger spaces.     

The radiation condition at infinity for the high-frequency Helmholtz equation with source term: a wave-packet approach

We consider the high-frequency Helmholtz equation with a given source term, and a small absorption parameter α>0. The high-frequency (or: semi-classical) parameter is ?>0. We let ? and α go to zero simultaneously. We assume that the zero energy is non-trapping for the underlying classical flow. We also assume that the classical trajectories starting from the origin satisfy a transversality condition, a generic assumption.Under these assumptions, we prove that the solution u^? radiates in the outgoing direction, uniformly in ?. In particular, the function u^?, when conveniently rescaled at the scale ? close to the origin, is shown to converge towards the outgoing solution of the Helmholtz equation, with coefficients frozen at the origin. This provides a uniform version (in ?) of the limiting absorption principle.Writing the resolvent of the Helmholtz equation as the integral in time of the associated semi-classical Schrödinger propagator, our analysis relies on the following tools: (i) for very large times, we prove and use a uniform version of the Egorov Theorem to estimate the time integral; (ii) for moderate times, we prove a uniform dispersive estimate that relies on a wave-packet approach, together with the above-mentioned transversality condition; (iii) for small times, we prove that the semi-classical Schrödinger operator with variable coefficients has the same dispersive properties as in the constant coefficients case, uniformly in ?.     

The Structure of the Closure of the Rational Functions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>(μ)

Let K be a compact subset in the complex plane and let A(K) be the uniform closure of the functions continuous on K and analytic on K°. Let μ be a positive finite measure with its support contained in K. For 1 ≤ q < ∞, let A^q(K, μ) denote the closure of A(K) in L^q(μ). The aim of this work is to study the structure of the space A^q(K, μ). We seek a necessary and sufficient condition on K so that a Thomson-type structure theorem for A^q(K, μ) can be established. Our theorem deduces J. Thomson’s structure theorem for P^q(μ), the closure of polynomials in L^q(μ), as the special case when K is a closed disk containing the support of μ.     

On the centralizer ofK in the universal enveloping algebra of SO(n,1) and SU(n,1)

LetG _o be a non compact real semisimple Lie group with finite center, and letU U(g) ^K denote the centralizer inU U(g) of a maximal compact subgroupK _o ofG _o. To study the algebraU U(g) ^K , B. Kostant suggested to consider the projection mapP:U U(g)→U(k)⊗U(a), associated to an Iwasawa decompositionG _o=K _o A _o N _o ofG _o, adapted toK _o. WhenP is restricted toU U(g) ^K J. Lepowsky showed thatP becomes an injective anti-homomorphism ofU U(g) ^K intoU(k) ^M ⊗U(a). HereU(k) ^M denotes the centralizer ofM _o inU(k),M _o being the centralizer ofA _o inK _o. To pursue this idea further it is necessary to have a good characterization of the image ofU U(g) ^K inU(k)^M×U(a). In this paper we describe such image whenG _o=SO(n,1)_e or SU(n,1). This is acomplished by establishing a (minimal) set of equations satisfied by the elements in the image ofU U(g) ^K , and then proving that they are enough to characterize such image. These equations are derived on one hand from the intertwining relations among the principal series representations ofG _o given by the Kunze-Stein interwining operators, and on the other hand from certain imbeddings among Verma modules. This approach should prove to be useful to attack the general case. Supported in part by Fundación Antorchas