Boundary Behavior Of Functions Research Articles

Sobolev spaces on p.c.f. self-similar sets II: Boundary behavior and interpolation theorems

Abstract We study the Sobolev spaces H σ ⁢ ( K ) {H^{\sigma}(K)} and H 0 σ ⁢ ( K ) {H^{\sigma}_{0}(K)} on p.c.f. self-similar sets. First, for σ ∈ ℝ + {\sigma\in\mathbb{R}^{+}} , we make an exact description of the tangents of functions in H σ ⁢ ( K ) {H^{\sigma}(K)} at the boundary, and introduce a countable set of critical orders that arises naturally in the boundary behavior of functions. These critical orders are just 1 2 + ℤ + {\frac{1}{2}+\mathbb{Z}_{+}} in the Euclidean case, but become complicated on fractals. Second, we characterize H 0 σ ⁢ ( K ) {H_{0}^{\sigma}(K)} as the space of functions in H σ ⁢ ( K ) {H^{\sigma}(K)} with tangents of appropriate order, that depend on σ and critical orders, being 0. Last, we extend H σ ⁢ ( K ) {H^{\sigma}(K)} to σ ∈ ℝ {\sigma\in\mathbb{R}} , and obtain various interpolation theorems with σ ∈ ℝ + {\sigma\in\mathbb{R}^{+}} or σ ∈ ℝ {\sigma\in\mathbb{R}} . The interpolation space presents a critical phenomenon when the resulted order σ θ {\sigma_{\theta}} is critical. Moreover, for the interpolation couple ( H 0 σ ⁢ ( K ) , H 0 σ ′ ⁢ ( K ) ) {(H^{\sigma}_{0}(K),H^{\sigma^{\prime}}_{0}(K))} , more than the classical theorem, our interpolation theorem fully covers the teratological case that { σ , σ ′ } {\{\sigma,\sigma^{\prime}\}} contains at least one critical order.

Caratheodory theorem about prime ends on Riemann surfaces

The present paper is a continuation of our research that was devoted to the theory of the boundary behavior of mappings on Riemann surfaces. Here we develop the theory of the boundary behavior of the mappings in the class FLD (mappings with finite length distortion) first introduced for the Euclidean spaces in the article of Martio--Ryazanov--Srebro--Yakubov at 2004 and then included in the known monograph of these authors in the modern mapping theory at 2009. As it was shown in the recent papers of Kovtonyuk-Petkov-Ryazanov at 2017, such mappings, generally speaking, are not mappings in the Sobolev classes because their first partial derivatives can be not locally integrable. At the same time, this class is a natural generalization of the well-known significant classes of isometries and quasi--isometries. We obtain here a series of criteria in terms of dilatations for the homeomorphic extension of the mappings with finite length distortion between domains on Riemann surfaces to the completions of the domains by prime ends of Caratheodory. Here we start from the general criterion in Lemma 1 in terms of singular functional parameters and then derive on this basis many other criteria. In particular, Lemma 1 implies Theorem 1 with a criterion of the Lehto type and Corollary 1 shows that the conclusion holds, if the dilatation grows not quickly than logarithm of the hyperbolic distance at every boundary point. The next consequence in Theorem 2 gives an integral criterion of the Orlicz type and Corollary 2 says on simple integral conditions of the exponential type. Further, Theorem 3 and Remark 2 contain criteria in terms of singular integrals of the Calderon--Zygmund type. The other criterion in Theorem 4 is the existence of a dominant for the dilatation in the class FMO (functions with finite mean oscillation), i.e., having a finite mean deviation from its mean value over infinitesimal discs centered at boundary points. In other words, the latter means that such a dominant has a finite dispersion over the given infinitesimal discs. In particular, the latter leads to Corollary 3 on a dominant in the well--known class BMO (bounded mean oscillation) by John--Nirenberg and to a simple criterion in Corollary 4 on finiteness of the average of the dilatation over infinitesimal disks centered at boundary points.

On nonhomeomorphic mappings with the inverse Poletsky inequality

The article investigates the local and boundary behavior of mappings with branching that satisfy the inverse inequality of the Poletsky type. It is proved that mappings of this type are logarithmically Hölder-continuous under the condition that the function Q responsible for a distortion of the modulus of the families of curves is integrable. A continuous extension of the indicated mappings to the boundary is obtained. In addition, the conditions under which the families of mentioned mappings are equicontinuous inside the domain and at its boundary are studied.

On the local and boundary behaviour of mappings of factor spaces

In this article, we study mappings acting between domains of two factor spaces by certain groups of Möbius automorphisms of the unit ball that act discontinuously and do not have fixed points. For such mappings, we have established estimates for the distortion of the modulus of families of paths, which are similar to the well-known Poletsky and Väisälä inequalities. As applications, we have obtained several important results on the local and boundary behaviour of mappings.

On nonhomeomorphic mappings with the inverse Poletsky inequality

As known, the modulus method is one of the most powerful research tools in the theory of mappings. Distortion of modulus has an important role in the study of conformal and quasiconformal mappings, mappings with bounded and finite distortion, mappings with finite length distortion, etc. In particular, an important fact is the lower distortion of the modulus under mappings. Such relations are called inverse Poletsky inequalities and are one of the main objects of our study. The use of these inequalities is fully justified by the fact that the inverse inequality of Poletsky is a direct (upper) inequality for the inverse mappings, if there exist. If the mapping has a bounded distortion, then the corresponding majorant in inverse Poletsky inequality is equal to the product of the maximum multiplicity of the mapping on its dilatation. For more general classes of mappings, a similar majorant is equal to the sum of the values of outer dilatations over all preimages of the fixed point. It the class of quasiconformal mappings there is no significance between the inverse and direct inequalities of Poletsky, since the upper distortion of the modulus implies the corresponding below distortion and vice versa. The situation significantly changes for mappings with unbounded characteristics, for which the corresponding fact does not hold. The most important case investigated in this paper refers to the situation when the mappings have an unbounded dilatation. The article investigates the local and boundary behavior of mappings with branching that satisfy the inverse inequality of Poletsky with some integrable majorant. It is proved that mappings of this type are logarithmically Holder continuous at each inner point of the domain. Note that the Holder continuity is slightly weaker than the classical Holder continuity, which holds for quasiconformal mappings. Simple examples show that mappings of finite distortion are not Lipschitz continuous even under bounded dilatation. Another subject of research of the article is boundary behavior of mappings. In particular, a continuous extension of the mappings with the inverse Poletsky inequality is obtained. In addition, we obtained the conditions under which the families of these mappings are equicontinuous inside and at the boundary of the domain. Several cases are considered: when the preimage of a fixed continuum under mappings is separated from the boundary, and when the mappings satisfy normalization conditions. The text contains a significant number of examples that demonstrate the novelty and content of the results. In particular, examples of mappings with branching that satisfy the inverse Poletsky inequality, have unbounded characteristics, and for which the statements of the basic theorems are satisfied, are given.

Mappings with finite length distortion and prime ends on Riemann surfaces

The present paper is a continuation of our research that was devoted to the theory of the boundary behavior of mappings in the Sobolev classes (mappings with generalized derivatives) on Riemann surfaces. Here we develop the theory of the boundary behavior of the mappings in the class of FLD (mappings with finite length distortion) first introduced for the Euclidean spaces in the article of Martio-Ryazanov-Srebro-Yakubov at 2004 and then included in the known book of these authors at 2009 on the modern mapping theory. As was shown in the recent papers of Kovtonyuk-Petkov-Ryazanov at 2017, such mappings, generally speaking, are not mappings in the Sobolev classes, because their first partial derivatives can be not locally integrable. At the same time, this class is a natural generalization of the well-known significant classes of isometries and quasiisometries. We prove here a series of criteria in terms of dilatations for the continuous and homeomorphic extensions to the boundary of the mappings with finite length distortion between domains on Riemann surfaces by Caratheodory prime ends. The criterion for the continuous extension of the inverse mapping to the boundary is turned out to be the very simple condition on the integrability of the dilatations in the first power. The criteria for the continuous extension of the direct mappings to the boundary have a much more refined nature. One of such criteria is the existence of a majorant for the dilatation in the class of functions with finite mean oscillation, i.e., having a finite mean deviation from its mean value over infinitesimal disks centered at boundary points. As consequences, the corresponding criteria for a homeomorphic extension of mappings with finite length distortion to the closures of domains by Caratheodory prime ends are obtained.

On mappings with finite length distortion on Riemann surfaces

The present paper is a natural continuation of our previous paper (2017) on the boundary behavior of mappings in the Sobolev classes on Riemann surfaces, where the reader will be able to find the corresponding historic comments and a discussion of many definitions and relevant results. The given paper was devoted to the theory of the boundary behavior of mappings with finite distortion by Iwaniec on Riemannian surfaces first introduced for the plane in the paper of Iwaniec T. and Sverak V. (1993) On mappings with integrable dilatation and then extended to the spatial case in the monograph of Iwaniec T. and Martin G. (2001) devoted to Geometric function theory and non-linear analysis. At the present paper, it is developed the theory of the boundary behavior of the so--called mappings with finite length distortion first introduced in the paper of Martio O., Ryazanov V., Srebro U. and Yakubov~E. (2004) in the spatial case, see also Chapter 8 in their monograph (2009) on Moduli in modern mapping theory. As it was shown in the paper of Kovtonyuk D., Petkov I. and Ryazanov V. (2017) On the boundary behavior of mappings with finite distortion in the plane, such mappings, generally speaking, are not mappings with finite distortion by Iwaniec because their first partial derivatives can be not locally integrable. At the same time, this class is a generalization of the known class of mappings with bounded distortion by Martio--Vaisala from their paper (1988). Moreover, this class contains as a subclass the so-called finitely bi-Lipschitz mappings introduced for the spatial case in the paper of Kovtonyuk D. and Ryazanov V. (2011) On the boundary behavior of generalized quasi-isometries, that in turn are a natural generalization of the well-known classes of bi-Lipschitz mappings as well as isometries and quasi-isometries. In the research of the local and boundary behavior of mappings with finite length distortion in the spatial case, the key fact was that they satisfy some modulus inequalities which was a motivation for the consideration more wide classes of mappings, in particular, the Q-homeomorphisms (2005) and the mappings with finite area distortion (2008). Hence it is natural that under the research of mappings with finite length distortion on Riemann surfaces we start from establishing the corresponding modulus inequalities that are the main tool for us. On this basis, we prove here a series of criteria in terms of dilatations for the continuous and homeomorphic extension to the boundary of the mappings with finite length distortion between domains on arbitrary Riemann surfaces.

On behavior of inverse homeomorphisms in terms of prime ends

As is known, even conformal mappings of plane simply connected domains do not, generally speaking, have continuous boundary extension in the Euclidean sense. One of the minimum requirements necessary for such an extension is the local connectedness of the definition domain of the corresponding map on its boundary. Of course, quite a lot of simply connected domains do not have this property. For example, the unit disk with a cut along the positive part of the real axis is not locally connected at the boundary. In the same way, the mapped domain must also satisfy certain conditions necessary for the continuous extension of a mapping. The situation changes significantly if we are not talking about the Euclidean boundary behavior of mappings, but about extension in terms of the so-called prime ends. In this case, the domain of definition of mappings should be only regular, that is, this domain should be the image of a domain with a locally quasiconformal boundary under some quasiconformal mapping. A similar requirement also applies to the mapped domain. In this article, we study the equicontinuous families of maps at inner and boundary points in the case where the prime ends of the domain serve as boundary points. Relatively speaking, the paper consists of two parts, one of which contains a number of auxiliary statements, and the second, the final part of the work, contains the formulation of the main theorem and its proof. We consider a class of homeomorphisms of Euclidean space, inverse of which distort moduli of families of paths by the Poletsky type inequality. Note that these classes include most well-known mappings, such as conformal mappings, quasiconformal mappings, mappings with finite length and area distortion, and so on. It should be noted that under conformal mappings, distortion of the modulus of families of paths does not occur, therefore, when passing to inverse mappings, we remain in the class under study. A similar situation is in the case of quasiconformal mappings, since, as is known, the inverse mapping to a quasiconformal is also quasiconformal. In more general situations, the studied configurations can turn out to be much more complicated, in particular, the transition to inverse mappings can significantly change their properties (this is confirmed by specific examples of mappings, which are rather easy to construct in this case). This article is actually devoted to the study of this particular case, that is, when we are dealing with a certain family of homeomorphisms with an unbounded characteristic, in addition, mappings inverse to them are studied. In more detail, we consider mappings whose inverse satisfy the upper distortion estimate of the modulus of families of paths with integrable majorant. In the article, we proved that the families of the indicated mappings are equicontinuous both at the inner and boundary points of the domain, provided that the majorant responsible for the distortion of the modulus of the families of paths is integrable, besides that, the definition and mapped domains are regular, and the boundary points are prime ends of the definition domain. The results obtained in the paper are applicable to well-known classes of mappings, such as mappings with bounded and finite distortion, as well as to the Sobolev and Orlicz-Sobolev classes.

On boundary behavior of mappings with two normalized conditions

On boundary behavior of mappings with two normalized conditions

Local and boundary behavior of maps in metric spaces

Local and boundary behavior of maps in metric spaces

Schwarz Problems for Poly-Hardy Space on the Unit Ball

In this paper we study the Schwarz boundary value problems \({\big(}\) for short BVP\({\big)}\) for the poly-Hardy space defined on the unit ball of higher dimensional Euclidean space \({\mathbb{R}^n}\) . We first discuss the boundary behavior of functions belonging to the poly-Hardy class. Then we construct the Schwarz kernel function, and describe the boundary properties of the Schwarz-type integrable operator. Finally, we study the Schwarz BVPs for the Hardy class and the poly-Hardy class on the unit ball of higher dimensional Euclidean space \({\mathbb{R}^n}\), and obtain the expressions of solutions, explicitly. As an application, the monogenic signals considered for the Hardy spaces defined on the unit sphere are reconstructed when the scalar- and sub-algebra-valued data are given, which is the extension of the analytic signals for the Hardy spaces on the unit circle of the complex plane.

On the boundary behavior of mappings with finite distortion in the plane

In terms of prime ends by Caratheodory, it is studied the boundary behavior of the socalled lower Q-homeomorphisms in the plane that are a natural generalization of the quasiconformal mappings. In particular, it was found a series of effective conditions on the function Q(z) for a homeomorphic extension of the given mappings to the boundary by prime ends. The developed theory is applied to mappings with finite distortion by Iwaniec as well as to finitely bi–Lipschitz mappings that a far–reaching extension of the known classes of isometric and quasiisometric mappings.

On the Boundary Behavior of Mappings in the Class $$W^{1,1}_\mathrm{loc}$$ W loc 1 , 1 on Riemann Surfaces

In terms of dilatations, it is proved a series of criteria for continuous and homeomorphic extension to the boundary of mappings with finite distortion between regular domains on the Riemann surfaces.

Граничное поведение отображений на финслеровых пространствах

Граничное поведение отображений на финслеровых пространствах

Boundary behavior of function from the Bourgain algebra {(\psi A(\bar{D}),L^\infty(D))}_b

Boundary behavior of function from the Bourgain algebra {(\psi A(\bar{D}),L^\infty(D))}_b

Boundary behavior of mappings in λ(ε)-regular metric spaces

The problem of extension of ring Q-homeomorphisms to the boundary between domains in λ(e)-regular metric spaces is investigated. The conditions imposed on the function Q(x) and the boundaries of domains under which every ring Q-homeomorphism admits a continuous or homeomorphic extension to the boundary are formulated.

Boundary behavior of functions from Sobolev classes defined on domains with exterior peak

We establish an invertible characteristic of the boundary behavior of functions from Sobolev spaces defined on a space domain having a vertex of exterior peak on the boundary. The boundary is assumed sufficiently smooth in a neighborhood of the peak vertex. The description of the traces on the boundary is given with the use of weighted Besov spaces.

Boundary behaviour of functions which possess universal Taylor series

It is known that, for any simply connected proper subdomain Ω of the complex plane and any point ζ in Ω, there are holomorphic functions on Ω that possess “universal”Taylor series expansions about ζ; that is, partial sums of the Taylor series approximate arbitrary polynomials on arbitrary compacta in C\Ω that have connected complement. This paper establishes a strong unboundedness property for such functions near every boundary point. The result is new even in the case of the disc, where it strengthens work of several authors.

The boundary behavior of functions with vanishing integrals over hyperbolic disks

The boundary behavior of functions with vanishing integrals over hyperbolic disks

Some Remarks on the Boundary Behaviors of Functions in the Monogenic Hardy Spaces

Some estimates and boundary properties for functions in the monogenic Hardy spaces are given.