The Dirichlet-type space ) is the Banach space of functions analytic in the unit disc with derivatives belonging to the Bergman space . Let be an analytic self-map of the disc and define for . The operator is bounded (respectively, compact) if and only if a related measure is Carleson (respectively, compact Carleson). If is bounded (or compact) on , then the same behavior holds on ) and on the weighted Dirichlet space . Compactness on implies that is compact on the Hardy spaces and the angular derivative exists nowhere on the unit circle. Conditions are given which, together with the angular derivative condition, imply compactness on the space . Inner functions which induce bounded composition operators on are discussed briefly.
Let be the unit disk. We show that for some relatively closed set there is a function that can be uniformly approximated on by functions of , but such that cannot be written as , with and uniformly continuous on . This answers a question of Stray.
It is shown that a set-valued mapping of a hyperconvex metric space which takes values in the space of nonempty externally hyperconvex subsets of always has a lipschitzian single valued selection which satisfies for all . (Here denotes the usual Hausdorff distance.) This fact is used to show that the space of all bounded -lipschitzian self-mappings of is itself hyperconvex. Several related results are also obtained.
We give a geometric characterization of those positive finite measures on with the upper density finite at -almost every , such that the principal value of the Cauchy integral of ,
exists for -almost all . This characterization is given in terms of the curvature of the measure . In particular, we get that for , -measurable (where is the Hausdorff -dimensional measure) with , if the principal value of the Cauchy integral of exists -almost everywhere in , then is rectifiable.
On bounded domains we consider the anisotropic problems in with 1$"> and on and in with and on . Moreover, we generalize these boundary value problems to space-dimensions 2$">. Under geometric conditions on and monotonicity assumption on we prove existence and uniqueness of positive solutions. 相似文献
The vanishing of Van Kampen's obstruction is known to be necessary and sufficient for embeddability of a simplicial -complex into for , and it was recently shown to be incomplete for . We use algebraic-topological invariants of four-manifolds with boundary to introduce a sequence of higher embedding obstructions for a class of -complexes in .
For a bounded invertible operator on a complex Banach space let be the set of operators in for which Suppose that and is in A bound is given on in terms of the spectral radius of the commutator. Replacing the condition in by the weaker condition as for every 0$">, an extension of the Deddens-Stampfli-Williams results on the commutant of is given.
Let , be a sequence of bounded pseudoconvex domains that converges, in the sense of Boas, to a bounded domain . We show that if can be described locally as the graph of a continuous function in suitable coordinates for , then the Bergman kernel of converges to the Bergman kernel of uniformly on compact subsets of . 相似文献
We show that finite dimensional injective operator spaces are corners of finite dimensional -algebras .
Let be a vector lattice of real functions on a set with , and let be a linear positive functional on . Conditions are given which imply the representation , , for some bounded charge . As an application, for any bounded charge on a field , the dual of is shown to be isometrically isomorphic to a suitable space of bounded charges on . In addition, it is proved that, under one more assumption on , is the integral with respect to a -additive bounded charge.