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Том 229, № 5 (2018)
- Год: 2018
- Статей: 10
- URL: https://ogarev-online.ru/1072-3374/issue/view/14892
Article
Properties of Some Extensions of the Quadratic Form of the Vector Laplace Operator
Аннотация
We consider the action of the quadratic form of the Laplace operator and its extensions in subspaces of linear combinations of the “transverse” and “longitudinal” functions with the fixed orbital momentum with respect to the coordinate origin. In the statement of the problem, it is required that the extensions obtained, after the transfer back to the space of vector functions, can be represented as simple limit expressions with two coefficients. We study the behavior of these coefficients with respect to the initial choice of the linear subspace. Bibliography: 5 titles.
487-496
An Analog of the Hyperbolic Metric Generated by a Hilbert Space with the Schwarz–Pick Kernel
Аннотация
It is proved that a Hilbert function space on a set X with the Schwarz–Pick kernel (this is a wider class than the class of Hilbert spaces with the Nevanlinna–Pick kernel) generates a metric on the set X which is an analog of the hyperbolic metric in the unit disk. For a sequence satisfying an abstract Blaschke condition, it is proved that the associated infinite Blaschke product converges uniformly on any fixed bounded set and in the strong operator topology of the multiplier space. Bibliography: 8 titles.
497-505
Notes on the Codimension One Conjecture in the Operator Corona Theorem
Аннотация
Answering a question of S. R. Treil (2004), for every δ, 0 < δ < 1, we construct examples of contractions whose characteristic function F ∈ H∞(ε →ε∗) satisfies the conditions ‖F(z)x‖ ≥ δ‖x‖ and dim ε∗ ⊝ F(z)ε = 1 for every z ∈ ????, x ∈ ε, but is not left invertible. In addition, we show that the condition where is the trace class of operators, which is sufficient for the left invertibility of an operatorvalued function F satisfying the estimate ‖F(z)x‖ ≥ δ‖x‖ for every z ∈ ????, x ∈ ε, with some δ > 0 (Treil, 2004), is necessary for the left invertibility of an inner function F such that dim ε∗ ⊝ F(z)ε < ∞ for some z ∈ ????.
506-517
518-527
528-533
Comparison of Boundary Smoothness for an Analytic Function and for its Modulus in the Case of the Upper Half-Plane
Аннотация
Results of a recent paper by A. V. Vasin, S. V. Kislyakov, and the author are extended to the case of outer functions in the upper half-plane. As in the case of the disk, it can only be guaranteed that the smoothness of an outer function is at least one half of that for its modulus, but the quantitative manifestation of this effect is different; in particular, it depends on the position of the point at which the smoothness is measured. Bibliography: 8 titles.
534-544
Generators for Spaces of Entire Functions with a System of Weighted Estimates
Аннотация
We consider spaces of entire functions with systems of weighted estimates. The case of twoterm weight sequences consisting of radial and nonradial components is studied. Under some assumptions on the weight sequence, we obtain a complete description of generators in these spaces. We apply this result to the problem of normal solvability of systems of convolution equations in the Roumieu spaces of ultradifferentiable functions and, as a particular case, in Gevrey classes.
545-560
A1-Regularity and Boundedness of Riesz Transforms in Banach Lattices of Measurable Functions
Аннотация
Let X be a Banach lattice of measurable functions on ℝn × Ω having the Fatou property. We show that the boundedness of all Riesz transforms Rj in X is equivalent to the boundedness of the Hardy–Littlewood maximal operator M in both X and X′, and thus to the boundedness of all Calderón–Zygmund operators in X. We also prove a result for the case of operators between lattices: If Y ⊃ X is a Banach lattice with the Fatou property such that the maximal operator is bounded in Y ′, then the boundedness of all Riesz transforms from X to Y is equivalent to the boundedness of the maximal operator from X to Y , and thus to the boundedness of all Calderón–Zygmund operators from X to Y .
561-567
Smoothness of a Holomorphic Function in a Ball and of its Modulus on the Sphere
Аннотация
Let a function f be holomorphic in the unit ball ????n, continuous in the closed ball \( {\overline{\mathbb{B}}}^n \) , and let f(z) ≠ 0, z ∈ ????n. Assume that |f| belongs to the α-Hölder class on the unit sphere Sn, 0 < α ≤ 1. The present paper is devoted to the proof of the statement that f belongs to the α/2-Hölder class on \( {\overline{\mathbb{B}}}^n \).
568-571
Exponential Synthesis in the Kernel of a Symmetric Convolution
Аннотация
The paper describes a certain class of homogeneous equations of convolution type in spaces of analytic functions on convex domains. We obtain sufficient conditions under which every solution of an equation from this class is approximated by its elementary solutions.
572-599