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Том 215, № 5 (2016)
- Год: 2016
- Статей: 11
- URL: https://ogarev-online.ru/1072-3374/issue/view/14744
Article
Commutator Lipschitz Functions and Analytic Continuation
Аннотация
Let \( \mathfrak{F} \)0 and \( \mathfrak{F} \) be perfect subsets of the complex plane ℂ. Assume that \( \mathfrak{F} \)0 ⊂ \( \mathfrak{F} \) and the set \( \Omega \overset{\mathrm{def}}{=}\mathfrak{F}\backslash {\mathfrak{F}}_0 \) is open. We say that a continuous function f : \( \mathfrak{F} \) → ℂ is an analytic continuation of a function f0 : \( \mathfrak{F} \)0 → ℂ if f is analytic on Ω and f|\( \mathfrak{F} \)0 = f0. In the paper, it is proved that if \( \mathfrak{F} \) is bounded, then the commutator Lipschitz seminorm of the analytic continuation f coincides with the commutator Lipschitz seminorm of f0. The same is true for unbounded \( \mathfrak{F} \) if some natural restrictions concerning the behavior of f at infinity are imposed.
543-551
552-559
Properties of the Radial Part of the Laplace Operator for l=1 in a Special Scalar Product
Аннотация
We develop self-adjoint extensions of the radial part of the Laplace operator for l = 1 in a special scalar product. The product arises under the passage of the standard product from ℝ3 to the set of functions parametrizing one of two components of the transverse vector field. Similar extensions are treated for the square of the inverse operator of the radial part in question. Bibliography: 8 titles.
560-573
574-576
Regularity of the Beurling Transform in Smooth Domains
Аннотация
The relationship between smoothness properties of the boundary of a domain Ω and the boundedness of the Beurling transform in the corresponding Lipschitz classes Lip(ω) for the case of a Dini-regular module of continuity ω is studied. The result is sharp. Our motivation arises from the work of Mateu, Orobitg, and Verdera.
577-584
Blaschke Product for a Hilbert Space with Schwarz–Pick Kernel
Аннотация
For an analog of a Blaschke product for a Hilbert space with Schwarz–Pick kernel (this is a wider class than the class of Hilbert spaces with Nevanlinna–Pick kernel), it is proved that only finitely many elementary multipliers may have zeros on a fixed compact set. It is also proved that partial Blaschke products multiplied by an appropriate reproducing kernel converge in the Hilbert space. These abstract theorems are applied to the weighted Hardy spaces in the unit disk and to the Drury–Arveson spaces. Bibliography: 11 titles.
585-594
Sharp Bernstein Type Inequalities for Splines in the Mean Square Metrics
Аннотация
We give an elementary proof of the sharp Bernstein type inequality
595-600
On Univalence of Solutions of Second-Order Elliptic Equations in the Unit Disk on the Plane
Аннотация
Sufficient conditions on a function continuous on the unit circle are found ensuring that the solution of the Dirichlet problem in the unit disk for a certain second-order partial differential equation with this boundary function is a homeomorphism of the unit disk onto a simply connected Jordan domain. Bibliography: 5 titles.
601-607
Drop of the Smoothness of an Outer Function Compared to the Smoothness of its Modulus, Under Restrictions on the Size of Boundary Values
Аннотация
Let F be an outer function on the unit disk. It is well known that its smoothness properties can be twice worse than those of the modulus of its boundary values, but under some restrictions on log |F|, this gap becomes smaller. It is shown that the smoothness decay admits a convenient description in terms of a rearrangement invariant Banach function space containing log |F|. All the results are of pointwise nature.
608-616
Summation Methods for Fourier Series with Respect to the Azoff–Shehada System
Аннотация
A special class of complete minimal systems with complete biorthogonal system in a Hilbert space is considered. This class was introduced by Azoff and Shehada. The paper studies conditions under which there exists a linear summation method for Fourier series with respect to the Azoff–Shehada system. A construction of a linear summation method of the Fourier series for a given vector is presented, as well as a construction of a universal linear summation method.
617-623
624-630