Embedding of Sobolev space in Orlicz space for a domain with irregular boundary

In this paper, we establish the embedding of a weighted Sobolev space in an Orlicz space for a domain with irregular boundary. We find an estimate of the order of growth of the N-function (defining the Orlicz space) and show that, under certain additional constraints on the weights, this estimate is sharp. We also establish the embedding in the space of continuous functions.     

Complexity spaces as quantitative domains of computation

We study domain theoretic properties of complexity spaces. Although the so-called complexity space is not a domain for the usual pointwise order, we show that, however, each pointed complexity space is an ω-continuous domain for which the complexity quasi-metric induces the Scott topology, and the supremum metric induces the Lawson topology. Hence, each pointed complexity space is both a quantifiable domain in the sense of M. Schellekens and a quantitative domain in the sense of P. Waszkiewicz, via the partial metric induced by the complexity quasi-metric.     

Norms of multiplication operators on Hardy spaces and weighted composition operators from Hardy spaces to weighted-type spaces on bounded symmetric domains

Let D be a bounded symmetric domain. We calculate operator norm of the multiplication operator on the Hardy space H^p(D), as well as of the weighted composition operator from H^p(D) to a weighted-type space.     

A discrete norm on a Lipschitz surface and the Sobolev straightening of a boundary

Let a piece of the boundary of a Lipschitz domain be parameterized conventionally and let the traces of functions in the Sobolev space W _p ² be written out through this parameter. In this space, we propose a discrete (diadic) norm generalizing A. Kamont’s norm in the plane case. We study the conditions when the space of traces coincides with the corresponding space for the plane boundary.     

A remark on the mean curvature of a graph-like hypersurface in hyperbolic space

In this paper we investigate the mean curvature H of a radial graph in hyperbolic space Hn+1. We obtain an integral inequality for H, and find that the lower limit of H at infinity is less than or equal to 1 and the upper limit of H at infinity is more than or equal to −1. As a byproduct we get a relation between the n-dimensional volume of a bounded domain in an n-dimensional hyperbolic space and the (n−1)-dimensional volume of its boundary. We also sharpen the main result of a paper by P.-A. Nitsche dealing with the existence and uniqueness of graph-like prescribed mean curvature hypersurfaces in hyperbolic space.     

TOEPLITZ OPERATORS ON HEISENBERG GROUP

Denote by Ω the Siegel domain in Cn, n 〉 1. In this paper, we study the essential spectra of Toeplitz operators defined on the Hardy space H2（а↓Ω）. In addition, the characteristic equation of analytic Toeplitz operators iааs obtained.     

Orlicz-Sobolev extensions and measure density condition

We study the extension properties of Orlicz-Sobolev functions both in Euclidean spaces and in metric measure spaces equipped with a doubling measure. We show that a set E⊂R satisfying a measure density condition admits a bounded linear extension operator from the trace space W1,Ψ(Rn)E_| to W1,Ψ(Rn). Then we show that a domain, in which the Sobolev embedding theorem or a Poincaré-type inequality holds, satisfies the measure density condition. It follows that the existence of a bounded, possibly non-linear extension operator or even the surjectivity of the trace operator implies the measure density condition and hence the existence of a bounded linear extension operator.     

Composition operators on the Newton space

We investigate properties of composition operators C? on the Newton space (the Hilbert space of analytic functions which have the Newton polynomials as an orthonormal basis). We derive a formula for the entries of the matrix of C? with respect to the basis of Newton polynomials in terms of the value of the symbol ? at the non-negative integers. We also establish conditions on the symbol ? for boundedness, compactness, and self-adjointness of the induced composition operator C?. A key technique in obtaining these results is use of an isomorphism between the Newton space and the Hardy space via the Binomial Theorem.     

ESTIMATES ON EIGENVALUES FOR THE BIHARMONIC OPERATOR ON A BOUNDED DOMAIN IN H~n（-1）

In this paper, we consider eigenvalues of the Dirichlet biharmonic operator on a bounded domain in a hyperbolic space. We obtain universal bounds on the (k + 1)th eigenvalue in terms of the first kth eigenvalues independent of the domains.     

The Berezin transform and Laplace-Beltrami operator

We study the Berezin transform of bounded operators on the Bergman space on a bounded symmetric domain Ω in Cn. The invariance of range of the Berezin transform with respect to G=Aut(Ω), the automorphism group of biholomorphic maps on Ω, is derived based on the general framework on invariant symbolic calculi on symmetric domains established by Arazy and Upmeier. Moreover we show that as a smooth bounded function, the Berezin transform of any bounded operator is also bounded under the action of the algebra of invariant differential operators generated by the Laplace-Beltrami operator on the unit disk and even on the unit ball of higher dimensions.     

A Banach lattice approach to convergent integrably bounded set-valued martingales and their positive parts

Denote by cf(X) the set of all nonempty convex closed subsets of a separable Banach space X. Let (Ω,Σ,μ) be a complete probability space and denote by (L¹[Σ,cf(X)],Δ) the complete metric space of (equivalence classes of a.e. equal) integrably bounded cf(X)-valued functions. For any preassigned filtration (Σi), we describe the space of Δ-convergent integrably bounded cf(X)-valued martingales in terms of the Δ-closure of in L¹[Σ,cf(X)]. In particular, we provide a formula to calculate the join of two such martingales and the positive part of such a martingale. Our object is achieved by considering the more general setting of a near vector lattice (S,d), endowed with a Riesz metric d. By means of Rådström's embedding theorem for such spaces, a link is established between the space of convergent martingales in S and the space of convergent martingales in the Rådström completion R(S) of S. This link provides information about the former space of martingales, via known properties of measure-free martingales in Riesz normed vector lattices, applicable to R(S). We also apply our general results to the spaces of Δ-convergent ck(X)-valued martingales, where ck(X) denotes the set of all nonempty convex compact subsets of X.     

Growth estimates in the Hardy-Sobolev space of an annular domain with applications

We give an explicit estimate on the growth of functions in the Hardy-Sobolev space Hk,2(Gs) of an annulus. We apply this result, first, to find an upper bound on the rate of convergence of a recovery interpolation scheme in H1,2(Gs) with points located on the outer boundary of Gs. We also apply this result for the study of a geometric inverse problem, namely we derive an explicit upper bound on the area of an unknown cavity in a bounded planar domain from the difference of two electrostatic potentials measured on the boundary, when the cavity is present and when it is not.     

Decay rate of solutions of Navier-Stokes equations in 3-dimensional half space

In this article, we derive the L² spatial decay rate of a weak solution of the incompressible flow in the half space. When the half space (or some other fluid region with boundary) is concerned, pressure estimate is main obstacle since we do not have enough information of the pressure on the boundary. In this paper, we give an idea which does not require any pressure information on the boundary, and we hope that our idea could be applied to other problems such as exterior domain problem.     

The extension of functions of Sobolev classes beyond the boundary of the domain on Carnot groups

We prove the theorem on extension of the functions of the Sobolev space W _p ^l (Ω) which are defined on a bounded (ε, δ)-domain Ω in a two-step Carnot group beyond the boundary of the domain of definition. This theorem generalizes the well-known extension theorem by P. Jones for domains of the Euclidean space.     

Rectifiable sets and coarea formula for metric-valued mappings

We study Lipschitz mappings defined on an Hn-rectifiable metric space with values in an arbitrary metric space. We find necessary and sufficient conditions on the image and the preimage of a mapping for the validity of the coarea formula. As a consequence, we prove the coarea formula for some classes of mappings with Hk-σ-finite image. We also obtain a metric analog of the Implicit Function Theorem. All these results are extended to large classes of mappings with values in a metric space, including Sobolev mappings and BV-mappings.     

Carleson Type Measures for Harmonic Mixed Norm Spaces with Application to Toeplitz Operators

Let Ω be a bounded domain in Rnwith a smooth boundary, and let h p,q be the space of all harmonic functions with a finite mixed norm. The authors first obtain an equivalent norm on h p,q, with which the definition of Carleson type measures for h p,q is obtained. And also, the authors obtain the boundedness of the Bergman projection on h p,q which turns out the dual space of h p,q. As an application, the authors characterize the boundedness(and compactness) of Toeplitz operators T μ on h p,q for those positive finite Borel measures μ.     

Structure of modules over the stereotype algebra ℒ(X) of operators

It is well known that every module M over the algebra ?(X) of operators on a finite-dimensional space X can be represented as the tensor product of X by some vector space E, M ? = E ? X. We generalize this assertion to the case of topological modules by proving that if X is a stereotype space with the stereotype approximation property, then for each stereotype module M over the stereotype algebra ? (X) of operators on X there exists a unique (up to isomorphism) stereotype space E such that M lies between two natural stereotype tensor products of E by X, $E \circledast X \subseteq M \subseteq E \odot X.$ . As a corollary, we show that if X is a nuclear Fréchet space with a basis, then each Fréchet module M over the stereotype operator algebra ?(X) can be uniquely represented as the projective tensor product of X by some Fréchet space E, $M = E \widehat \otimes X$ .     

k-super-strongly convex and k-super-strongly smooth Banach spaces

In this paper, we introduce two types of new Banach spaces: k-super-strongly convex spaces and k-super-strongly smooth spaces. It is proved that these two notions are dual. We also prove that the class of k-super-strongly convexifiable spaces is strictly between locally k-uniformly rotund spaces and k-strongly convex spaces, and obtain some necessary and sufficient conditions of k-super-strongly convex space (respectively k-super-strongly smooth space). Also, for each k?2, it is shown that there exists a k-super-strongly convex (respectively k-super-strongly smooth) space which is not (k−1)-super-strongly convex (respectively (k−1)-super-strongly smooth) space.     

Weakly K-analytic spaces and the three-space property for analyticity

Let (E,E^′) be a dual pair of vector spaces. The paper studies general conditions which allow to lift analyticity (or K-analyticity) from the weak topology σ(E,E^′) to stronger ones in the frame of (E,E^′). First we show that the Mackey dual of a space Cp(X) is analytic iff the space X is countable. This yields that for uncountable analytic spaces X the Mackey dual of Cp(X) is weakly analytic but not analytic. We show that the Mackey dual E of an (LF)-space of a sequence of reflexive separable Fréchet spaces with the Heinrich density condition is analytic, i.e. E is a continuous image of the Polish space NN. This extends a result of Valdivia. We show also that weakly quasi-Suslin locally convex Baire spaces are metrizable and complete (this extends a result of De Wilde and Sunyach). We provide however a large class of weakly analytic but not analytic metrizable separable Baire topological vector spaces (not locally convex!). This will be used to prove that analyticity is not a three-space property but we show that a metrizable topological vector space E is analytic if E contains a complete locally convex analytic subspace F such that the quotient E/F is analytic. Several questions, remarks and examples are included.     

Normality in products with a countable factor

The class of normal spaces that have normal product with every countable space is considered. A countably compact normal space X and a countable Y such that X×Y is not normal is constructed assuming CH. Also, ? is used to construct a perfectly normal countably compact X and a countable Y such that X×Y is not normal. The question whether a Dowker space can have normal product with itself is considered. It is shown that if X is Dowker and contains any countable non-discrete subspace, then X² is not normal. It follows that a product of a Dowker space and a countable space is normal if and only if the countable space is discrete. If X is Rudin's ZFC Dowker space, then X² is normal. An example of a Dowker space of cardinality ℵ₂ with normal square is constructed assuming .