Hardy Spaces for Quasiregular Mappings and Composition Operators

We study Hardy spaces H p \mathcal {H}^p , 0 > p > ∞ 0>p>\infty for quasiregular mappings on the unit ball B B in R n {\mathbb R}^n which satisfy appropriate growth and multiplicity conditions. Under these conditions we recover several classical results for analytic functions and quasiconformal mappings in H p \mathcal {H}^p . In particular, we characterize H p \mathcal {H}^p in terms of non-tangential limit functions and non-tangential maximal functions of quasiregular mappings. Among applications we show that every quasiregular map in our class belongs to H p \mathcal {H}^p for some p = p ( n , K ) p=p(n,K) . Moreover, we provide characterization of Carleson measures on B B via integral inequalities for quasiregular mappings on B B . We also discuss the Bergman spaces of quasiregular mappings and their relations to H p \mathcal {H}^p spaces and analyze correspondence between results for H p \mathcal {H}^p spaces and A \mathcal {A} -harmonic functions. A key difference between the previously known results for quasiconformal mappings in R n {\mathbb R}^n and our setting is the role of multiplicity conditions and the growth of mappings that need not be injective. Our paper extends results by Astala and Koskela, Jerison and Weitsman, Jones, Nolder, and Zinsmeister.

Here we consider the classes of mappings of metric spaces that distort the modulus of families of paths similarly to Poletsky inequality. For domains, which are not locally connected at the boundaries, we obtain results on the boundary extension of the indicated mappings. We also investigate the local and global behaviorof mappings in the context of the equicontinuity of their families. The main statements of the article are proved under the condition that the majorant responsible for the distortion of the modulus of the families of paths has a finite mean oscillation at the corresponding points. The results are applicable to well-known classes of conformal and quasiconformal mappings as well as mappings with a finite distortion.

Composition operators on the Hardy space ${H^2}$ of the ball in ${C^2}$ are studied. Some sufficient conditions are given for a composition operator to be bounded. A class of inner mappings is given which induces isometric composition operators. Another class of inner mappings is shown to induce unbounded composition operators.

We study images of the unit ball under certain special classes of quasiregular mappings. For homeomorphic, i.e., quasiconformal mappings problems of this type have been studied extensively in the literature. In this paper we also consider non-homeomorphic quasiregular mappings. In particular, we study (topologically) closed quasiregular mappings originating from the work of J. V\"ais\"al\"a and M. Vuorinen in 1970's. Such mappings need not be one-to-one but they still share many properties of quasiconformal mappings. The global behavior of closed quasiregular mappings is similar to the local behavior of quasiregular mappings restricted to a so-called normal domain.

As is known, the local behavior of maps is one of the most important problems of analysis. This, in particular, relates to the study of mappings with bounded and finite distortion, which have been actively studied recently. As for this work, here we solve the problem of the behavior of maps, the inverse of which satisfies the Poletsky inequality. The main result is the statement about the equicontinuity of the indicated mappings inside the domain in the case when the majorant corresponding to the distortion of the module under the mapping is integrable in the original domain. It should be emphasized that the proof of this result is largely geometric, at the same time, it uses only the conditions of boundedness of the direct and mapped domains and does not involve any requirements on their boundaries. The study of families of mappings inverse to a given class may turn out to be trivial if we are talking about quasiconformal mappings. In the latter case, we do not go beyond the limits of the class under study in the transition to inverse maps. Nevertheless, when studying mappings with unbounded characteristic, this question is quite substantial, as simple examples of the corresponding classes show. The idea of the proof of the main result is based on the fact that the inner points of an arbitrary domain are weakly flat. The last statement can be called the Väisälä lemma, which was established in his monograph and related to families of curves joining two continua between the plates of a spherical condenser. The proof is also based on the fact that the module of families of curves joining two converging continua in a good domain must tend to infinity. In this case, the neighborhood of some inner point of the mapped domain serves as ''good'' region, in which we check the equicontinuity of the inverse family of maps. The results of this article are applicable to many other classes of mappings such as mappings with a finite distortion in the sense of Iwaniec, Sobolev classes on the plane and in space, and so on.

<p>We consider a wider class of nonexpansive type mappings and present some fixed point results for this class of mappingss in hyperbolic spaces. Indeed, first we obtain some existence results for this class of mappings. Next, we present some convergence results for an iteration algorithm for the same class of mappings. Some illustrative non-trivial examples have also been discussed.</p>

We consider a wider class of nonexpansive type mappings and present some fixed point results for this class of mapping in partially hyperbolic spaces. Indeed, first we obtain some existence results for this class of mappings. Next, we present some convergence results for an iteration algorithm for the same class of mappings. Some illustrative nontrivial examples have also been discussed. Finally, we provide an application of our results to nonlinear integral equations.

A model that describes periodic changes in the lemming population size as a function of the physical characteristics of ecologically significant parameters of the habitat is analyzed. Earlier studies on modeling of the population size dynamics for tundra animals substantiated a new class of one-dimensional unimodal mappings of a state space into itself. The dependence of trajectory stability on parameter variation and the bifurcation and asymptotic properties of the trajectories are in the focus of attention when this class of mappings is studied. Methods for approximating implicitly defined sets are used for numerical analysis of the maps: trajectory tubes, attractors, and bifurcation diagrams are constructed. An example of a bifurcation diagram construction along the contour in the parameter space is given for this class of mappings; the diagram enables the analysis of the dependence of trajectory behavior on changes of the modeling map parameters. Thus, the use of the mapping of the type considered for the model description of the lemming population enabled the study of the effect of long-term biospheric rhythms, which include time intervals with extreme living conditions. The study showed that such impacts do not cause degeneration of the population: the periods with chaotic dynamics that emerge under extreme conditions are replaced by well-ordered behavior in the form of small-period population cycles.

Two theorems are given regarding the means of quasiconformal and quasiregular mappings. Together they show that the principle of subordination for means of analytic functions has no analog, at least in the case of plane quasiregular mappings.

We are concerned with Hardy and BMO spaces of operator-valued functions analytic in the unit disk of C. In the case of the Hardy space, we involve the atomic decomposition since the usual argument in the scalar setting is not suitable. Several properties (the Garsia-norm equivalent theorem, Carleson measure, and so on) of BMOA spaces are extended to the operator-valued setting. Then, the operator-valued H-1-BMOA duality theorem is proved. Finally, by the H-1-BMOA duality we present the Lusin area integral and Littlewood-Paley g-function characterizations of the operator-valued analytic Hardy space.

Weak and strong convergence theorems for nonspreading-type mappings in Hilbert spaces

In this paper, we introduce a new class of contractive mappings in a fuzzy metric space and we present fixed point results for this class of maps.

The authors study mappings that satisfy some estimate of the distortion of the modulus of families of paths. Under certain conditions on the domains between which the mappings act, we established that, these mappings are Hölder logarithmic continuous at the boundary points. It is known that, the Hölder continuity is established for many classes of mappings, say quasiconformal and quasiregular mappings. In this regard, it is possible to point to the classical distortion estimates by Martio-Rickman-Väisälä type, as well as the estimates related to the modern classes of mappings with finite distortion. In particular, V.I. Ryazanov together with R.R. Salimov and E.O. Sevost'yanov established local distortion estimates for plane and spatial mappings under FMO condition, or under the Lehto-type integral condition. Recently, the second co-author have obtained Hölder logarithmic continuity for the studied class at points of the unit sphere. This article considers the situation of similar mappings of different domains, not only the unit sphere. Namely, we consider mappings between quasiextremal distance domains (QED-domains) and convex domains. Note that, quasiextremal distance domains introduced by Gehring and Martio are structures in which the modulus of families of paths is metrically related to the diameter of sets. Also, convex domains are involved in the formulation of the main result; we consider mappings that surjectively act onto them. In addition, the article contains the formulations and proofs for some other results on this topic. We consider several more cases in detail, in particular when: 1) the definition domain is a domain with a locally quasiconformal boundary, and the image domain is a bounded convex domain; 2) the definition domain is a regular domain in the sense of prime ends, and the image domain is a bounded convex domain; 3) the mapping acts between the QED-domain and the bounded convex domain and has a fixed point. In all three cases, the mapping is Hölder logarithmic continuous; moreover, in case 2), which refers to prime ends, logarithmic continuity should also be understood in terms of prime ends.

Quasiregular mappings are a natural generalization of analytic functions to higher dimensions. Quasiregular mappings have many properties. Our work in this paper is to prove the following theorem: If f  a b is a quasiregular mapping which maps the plane onto the plane, then f is a bijection. We do this by finding the connection between quasiregular and quasiconformal mappings.

We develop a new tool based on quasiconformal dynamics and conformal dynamics of discrete group actions in 3-geometries to construct new types of quasiregular and quasisymmetric mappings in space. This tool has close relations to new effects in Teichmüller spaces of conformally flat structures on closed hyperbolic 3-manifolds/orbifolds and non-trivial hyperbolic 4-cobordisms, to the hyperbolic and conformal interbreedings as well as to non-faithful discrete representations of uniform hyperbolic 3-lattices. We demonstrate several applications of this tool and new types of quasiregular mappings in space. Leaving such applications to geometry and topology of manifolds to another our papers [10, 11], here we continue a series of applications of our constructions to long standing problems for quasiregular mappings in space, including M.A. Lavrentiev surjectivity problem, Pierre Fatou problem on radial limits and Matti Vuorinen injectivity and asymptotics problems for bounded quasiregular mappings in the unit 3-ball (cf. [4, 7-9]).