Interpolating sequences in harmonically weighted Dirichlet spaces

We describe the interpolating sequences and weak interpolating sequences for the multiplier algebras of harmonically weighted Dirichlet spaces when is a finitely atomic measure.

     

Local interpolation in Hilbert spaces of Dirichlet series

We denote by the Hilbert space of ordinary Dirichlet series with square-summable coefficients. The main result is that a bounded sequence of points in the half-plane is an interpolating sequence for if and only if it is an interpolating sequence for the Hardy space of the same half-plane. Similar local results are obtained for Hilbert spaces of ordinary Dirichlet series that relate to Bergman and Dirichlet spaces of the half-plane .

     

Stabilization of Tsirelson-type norms on spaces

We consider classical Tsirelson-type norms of and their modified versions on spaces, . We show that the modified Tsirelson-type norms do not distort any of the subspaces of the spaces. We prove that Tsirelson-type norms, being equivalent to their modified versions, may at most 2-distort spaces.

     

A version of Burkholder's theorem for operator-weighted spaces

Let be an operator weight, i.e. a weight function taking values in the bounded linear operators on a Hilbert space . We prove that if the dyadic martingale transforms are uniformly bounded on for each dyadic grid in , then the Hilbert transform is bounded on as well, thus providing an analogue of Burkholder's theorem for operator-weighted -spaces. We also give a short new proof of Burkholder's theorem itself. Our proof is based on the decomposition of the Hilbert transform into ``dyadic shifts'.

     

estimate of a parabolic Monge-Ampère equation on

We consider a special type of parabolic Monge-Ampère equation on arising from convex hypersurfaces expansion in Euclidean spaces. We obtained a parabolic estimate of the support functions for the convex hypersurfaces assuming that we have already had a parabolic estimate.

     

Canonical divisors in weighted Bergman spaces

Canonical divisors in Bergman spaces can be found as solutions of extremal problems. We derive a formula for certain extremal functions in the weighted Bergman spaces for -1$"> and . This leads to a study of the zeros of a specific family of hypergeometric functions.     

Hardy inequalities with optimal constants and remainder terms

We show that the classical Hardy inequalities with optimal constants in the Sobolev spaces and in higher-order Sobolev spaces on a bounded domain can be refined by adding remainder terms which involve norms. In the higher-order case further norms with lower-order singular weights arise. The case being more involved requires a different technique and is developed only in the space .

     

Triplets of Closely Embedded Dirichlet Type Spaces on the Unit Polydisc

We propose a general concept of triplet of Hilbert spaces with closed embeddings, instead of continuous ones, and we show how rather general weighted $L^2$ spaces yield this kind of generalized triplets of Hilbert spaces for which the underlying spaces and operators can be explicitly calculated. Then we show that generalized triplets of Hilbert spaces with closed embeddings can be naturally associated to any pair of Dirichlet type spaces $\mathcal{D }_\alpha (\mathbb{D }^N)$ of holomorphic functions on the unit polydisc $\mathbb{D }^N$ and we explicitly calculate the associated operators in terms of reproducing kernels and radial derivative operators. We also point out a rigging of the Hardy space $H^2(\mathbb{D }^N)$ through a scale of Dirichlet type spaces and Bergman type spaces.     

Sequential product of quantum effects

Unsharp quantum measurements can be modelled by means of the class of positive contractions on a Hilbert space , in brief, quantum effects. For the operation of sequential product was proposed as a model for sequential quantum measurements. We continue these investigations on sequential product and answer positively the following question: the assumption implies .

Then we propose a geometric approach of quantum effects and their sequential product by means of contractively contained Hilbert spaces and operator ranges. This framework leads us naturally to consider lattice properties of quantum effects, sums and intersections, and to prove that the sequential product is left distributive with respect to the intersection.

     

Spaces of Dirichlet series with the complete Pick property

We consider reproducing kernel Hilbert spaces of Dirichlet series with kernels of the form \(k\left( {s,u} \right) = \sum {{a_n}} {n^{ - s - \overline u }}\), and characterize when such a space is a complete Pick space. We then discuss what it means for two reproducing kernel Hilbert spaces to be “the same”, and introduce a notion of weak isomorphism. Many of the spaces we consider turn out to be weakly isomorphic as reproducing kernel Hilbert spaces to the Drury–Arveson space H _d ² in d variables, where d can be any number in {1, 2,...,∞}, and in particular their multiplier algebras are unitarily equivalent to the multiplier algebra of H _d ² . Thus, a family of multiplier algebras of Dirichlet series is exhibited with the property that every complete Pick algebra is a quotient of each member of this family. Finally, we determine precisely when such a space of Dirichlet series is weakly isomorphic as a reproducing kernel Hilbert space to H _d ² and when its multiplier algebra is isometrically isomorphic to Mult(H _d ² ).     

Global well-posedness of the Benjamin-Ono equation in low-regularity spaces

We prove that the Benjamin-Ono initial-value problem is globally well-posed in the Banach spaces , , of real-valued Sobolev functions.

     

Morse-Palais lemma for nonsmooth functionals on normed spaces

Using elementary differential calculus we get a version of the Morse-Palais lemma. Since we do not use powerful tools in functional analysis such as the implicit theorem or flows and deformations in Banach spaces, our result does not require the -smoothness of functions nor the completeness of spaces. Therefore it is stronger than the classical one but its proof is very simple.

     

Toric hypersymplectic quotients

We study the hypersymplectic spaces obtained as quotients of flat hypersymplectic space by the action of a compact Abelian group. These -dimensional quotients carry a multi-Hamilitonian action of an -torus. The image of the hypersymplectic moment map for this torus action may be described by a configuration of solid cones in . We give precise conditions for smoothness and non-degeneracy of such quotients and show how some properties of the quotient geometry and topology are constrained by the combinatorics of the cone configurations. Examples are studied, including non-trivial structures on and metrics on complements of hypersurfaces in compact manifolds.

     

A Lipschitz estimate for Berezin's operator calculus

F. A. Berezin introduced a general ``symbol calculus" for linear operators on reproducing kernel Hilbert spaces. For the particular Hilbert space of Gaussian square-integrable entire functions on complex -space, , we obtain Lipschitz estimates for the Berezin symbols of arbitrary bounded operators. Additional properties of the Berezin symbol and extensions to more general reproducing kernel Hilbert spaces are discussed.

     

The Dirichlet problem on quadratic surfaces

We give a fast, exact algorithm for solving Dirichlet problems with polynomial boundary functions on quadratic surfaces in such as ellipsoids, elliptic cylinders, and paraboloids. To produce this algorithm, first we show that every polynomial in can be uniquely written as the sum of a harmonic function and a polynomial multiple of a quadratic function, thus extending a theorem of Ernst Fischer. We then use this decomposition to reduce the Dirichlet problem to a manageable system of linear equations. The algorithm requires differentiation of the boundary function, but no integration. We also show that the polynomial solution produced by our algorithm is the unique polynomial solution, even on unbounded domains such as elliptic cylinders and paraboloids.

     

-estimate for the discrete Plateau Problem

In this paper we prove the convergence rates for a fully discrete finite element procedure for approximating minimal, possibly unstable, surfaces.

Originally this problem was studied by G. Dziuk and J. Hutchinson. First they provided convergence rates in the and norms for the boundary integral method. Subsequently they obtained the convergence estimates using a fully discrete finite element method. We use the latter framework for our investigation.

     

Regularity estimates for elliptic boundary value problems in Besov spaces

We consider the Dirichlet problem for Poisson's equation on a nonconvex plane polygonal domain . New regularity estimates for its solution in terms of Besov and Sobolev norms of fractional order are proved. The analysis is based on new interpolation results and multilevel representations of norms on Sobolev and Besov spaces. The results can be extended to a large class of elliptic boundary value problems. Some new sharp finite element error estimates are deduced.

     

Global existence for energy critical waves in -d domains

We prove that the defocusing quintic wave equation, with Dirichlet boundary conditions, is globally well posed on for any smooth (compact) domain . The main ingredient in the proof is an spectral projector estimate, obtained recently by Smith and Sogge, combined with a precise study of the boundary value problem.

     

Hilbert transform of

There are two general ways to evaluate the Hilbert transform of a function of real variable . We can extend to a harmonic function in the upper half plane by the Poisson integral formula. Non-tangential limit of its harmonic conjugate exists almost everywhere and is defined to be the Hilbert transform of . There is also a singular integral formula for the Hilbert transform of . It is fairly difficult to directly evaluate the Hilbert transform of . In this paper we give an explicit formula for the Hilbert transform of , where is a function in the Cartwright class.

     

Projections in operator ranges

If is a Hilbert space, is a positive bounded linear operator on and is a closed subspace of , the relative position between and establishes a notion of compatibility. We show that the compatibility of is equivalent to the existence of a convenient orthogonal projection in the operator range with its canonical Hilbertian structure.