Hardy spaces and quasiregular mappings

The condition (1.1) means that for all D ~ G the coordinate functions o f f belong to the Sobolev space W,~(D) of functions in L"(D) whose distributional first order partial derivatives are also L" integrable in D. In the above definition f ' ( x ) is the formal derivative of f a t x defined by means of the partial derivatives, I f ' (x) l is the operator norm of f '(x), and Ji(x) is the Jacobian determinant of f a t x. These are uniquely defined almost everywhere by (1.1). It turns out that quasiregular mappings are differentiable almost everywhere and a nonconstant quasiregular map is discrete, open, and sense-preserving. For these and other basic properties of qr maps we refer to books [Re] by Reshetnyak, [Vu] by Vuorinen, and also to a forthcoming book [R6] by the second author. The definition extends immediately to the case f : M ~ N where M and N are connected oriented Riemannian n-manifolds. We say that a continuous map f: M ~ N is quasiregular if it has a local representation at each point of M which is quasiregular in the above sense, and if there exists K, 1 < K < oe, such that

We study, for the first time, the maximum modulus set of a quasiregular map. It is easy to see that these sets are necessarily closed, and contain at least one point of each modulus. Blumenthal showed that for entire maps these sets are either the whole plane, or a countable union of analytic curves. We show that in the quasiregular case, by way of contrast, any closed set containing at least one point of each modulus can be attained as the maximum modulus set of a quasiregular map. These examples are all of polynomial type. We also show that, subject to an additional constraint, such sets can even be attained by quasiregular maps of transcendental type.

We define a quasi-Fatou component of a quasiregular map as a connected component of the complement of the Julia set. A domain in ℝd is called hollow if it has a bounded complementary component. We show that for each d ⩾ 2 there exists a quasiregular map of transcendental type f: ℝd → ℝd with a quasi-Fatou component which is hollow.Suppose that U is a hollow quasi-Fatou component of a quasiregular map of transcendental type. We show that if U is bounded, then U has many properties in common with a multiply connected Fatou component of a transcendental entire function. On the other hand, we show that if U is not bounded, then it is completely invariant and has no unbounded boundary components. We show that this situation occurs if J(f) has an isolated point, or if J(f) is not equal to the boundary of the fast escaping set. Finally, we deduce that if J(f) has a bounded component, then all components of J(f) are bounded.

Suppose $$p\ge 1$$ , $$w=P[F]$$ is a harmonic mapping of the unit disk $${\mathbb D}$$ satisfying F is absolutely continuous and $${\dot{F}}\in L^p(0, 2\pi )$$ , where $${\dot{F}}(e^{it})=\frac{\mathrm {d}}{\mathrm {d}t}F(e^{it})$$ . In this paper, we obtain Bergman norm estimates of the partial derivatives for w, i.e., $$\Vert w_z\Vert _{L^p}$$ and $$\Vert \overline{w_{{\bar{z}}}}\Vert _{L^p}$$ , where $$1\le p<2$$ . Furthermore, if w is a harmonic quasiregular mapping of $${\mathbb {D}}$$ , then we show that $$w_z$$ and $$\overline{w_{{\bar{z}}}}$$ are in the Hardy space $$H^p$$ , where $$1\le p\le \infty $$ . The corresponding Hardy norm estimates, $$\Vert w_z\Vert _{p}$$ and $$\Vert \overline{w_{{\bar{z}}}}\Vert _{p}$$ , are also obtained.

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.

In this paper we discuss some basic properties of octonionic Bergman and Hardy spaces. In the first part we review some fundamental concepts of the general theory of octonionic Hardy and Bergman spaces together with related reproducing kernel functions in the monogenic setting. We explain how some of the fundamental problems in well-defining a reproducing kernel can be overcome in the non-associative setting by looking at the real part of an appropriately defined para-linear octonion-valued inner product. The presence of a weight factor of norm 1 in the definition of the inner product is an intrinsic new ingredient in the octonionic setting. Then we look at the slice monogenic octonionic setting using the classical complex book structure. We present explicit formulas for the slice monogenic reproducing kernels for the unit ball, the right octonionic half-space and strip domains bounded in the real direction. In the setting of the unit ball we present an explicit sequential characterization which can be obtained by applying the special Taylor series representation of the slice monogenic setting together with particular octonionic calculation rules that reflect the property of octonionic para-linearity.

The goal of this survey is to give the reader a brief overview of the theory of quasiconformal (qc) and quasiregular (qr) mappings and of some related topics.

This chapter takes up several independent topics in Euclidean Fourier analysis, all having some bearing on the subject of partial differential equations. Section 1 elaborates on the relationship between the Fourier transform and the Schwartz space, the subspace of $L^{1}(\mathbb{R}^{N})$ consisting of smooth functions with the property that the product of any iterated partial derivative of the function with any polynomial is bounded. It is possible to make the Schwartz space into a metric space, and then one can consider the space of continuous linear functionals; these continuous linear functionals are called “tempered distributions.” The Fourier transform carries the space of tempered distributions in one-one fashion onto itself. Section 2 concerns weak derivatives, and the main result is Sobolev’s Theorem, which tells how to recover information about ordinary derivatives from information about weak derivatives. Weak derivatives are easy to manipulate, and Sobolev’s Theorem is therefore a helpful tool for handling derivatives without continually having to check the validity of interchanges of limits. Sections 3–4 concern harmonic functions, those functions on open sets in Euclidean space that are annihilated by the Laplacian. The main results of Section 3 are a characterization of harmonic functions in terms of a mean-value property, a reflection principle that allows the extension to all of Euclidean space of any harmonic function in a half space that vanishes at the boundary, and a result of Liouville that the only bounded harmonic functions in all of Euclidean space are the constants. The main result of Section 4 is a converse to properties of Poisson integrals for half spaces, showing that harmonic functions in a half space are given as Poisson integrals of functions or of finite complex measures if their $L^{p}$ norms over translates of the bounding Euclidean space are bounded. Sections 5–6 concern the Calderon–Zygmund Theorem, a far-reaching generalization of the theorem concerning the boundedness of the Hilbert transform. Section 5 gives the statement and proof, and two applications are the subject of Section 6. One of the applications is to Riesz transforms, and the other is to the Beltrami equation, whose solutions are “quasiconformal mappings.” Sections 7–8 concern multiple Fourier series for smooth periodic functions. The theory is established in Section 7, and an application to traces of integral operators is given in Section 8.

Lengths, areas and Lipschitz-type spaces of planar harmonic mappings

We prove a quasiconformal analogue of Koebe’s theorem related to the average Jacobian and use a normal family argument here to prove a quasiregular analogue of this result in certain domains inn-dimensional space. As an application, we establish that Lipschitz-type properties are inherited by a quasiregular function from its modulo. We also prove some results of Hardy-Littlewood type for Lipschitz-type spaces in several dimensions, give the characterization of Lipschitz-type spaces for quasiregular mappings by the average Jacobian, and give a short review of the subject.

Recently many important results on rings and quasiconformal mappings in space have been obtained by B. V. Šabat [9], F. W. Gehring [3], J. Väisälä [11] and others. The modulus of a ring in space can be defined in three apparently different but essentially equivalent ways. (See Gehring [4]). In the theory of quasiconformal mappings in space, some properties for moduli of rings in space play an important role, because the method by means of moduli acts also as a substitute in space for the Riemann mapping theorem which does not hold in space.

In this paper we study the rigidity theorem for smooth Bakry-Emery-Kohn harmonic function u u in the unit ball B n B_n in C n \textit {C}^n , which satisfies ◻ ψ u = ∑ i , j = 1 n ( δ i j − z i z ¯ j ) ∂ 2 u ∂ z i ∂ z ¯ j + ( 1 − | z | 2 ) ψ ′ ( | z | 2 ) ∑ j = 1 n z ¯ j ∂ u ∂ z ¯ j = 0 \begin{equation*} \Box ^{\psi } u=\sum _{i, j=1}^n (\delta _{ij}-z_i \overline {z}_j){\partial ^2 u \over \partial z_i \partial \overline {z}_j}+(1-|z|^2) \psi ’(|z|^2) \sum _{j=1}^n \overline {z}_j {\partial u\over \partial \overline {z}_j}=0 \end{equation*} with some restriction of the coefficients of Taylor expansion for ψ \psi at 1 1 . We prove that any smooth B-E-K harmonic function on B ¯ n \overline {B}_n must be holomorphic in B n B_n . We study the regularity problem for the solution of the Dirichlet boundary value problem: { ◻ ψ u = 0 , if z ∈ B n , u = f , if z ∈ ∂ B n . \begin{equation*} \begin {cases} \Box ^\psi u=0, \hbox { if } z\in B_n,\cr \quad \ u=f, \hbox { if } z\in \partial B_n.\cr \end{cases} \end{equation*}

We prove the following. If f is a harmonic quasiconformal mapping between the unit ball in $\mathbb {R}^{n}$ and a spatial domain with C1,α boundary, then f is Lipschitz continuous in B. This generalizes some known results for n = 2 and improves some others in higher dimensional case.

We prove that !u( ) ≤ C!f ( ), where u : Ω → R n is the harmonic extension of a continuous map f : @Ω → R n , if u is a K-quasiregular map and Ω is bounded in R n with C 2 boundary. Here C is a constant depending only on n, !f and K and !h denotes the modulus of continuity of h. We also prove a version of this result for Λ!-extension domains with c-uniformly perfect boundary and quasiconformal mappings.

We define Hardy spaces of functions taking values on a Banach space X over nonsmooth domains. The types of functions we consider are harmonic functions on a starlike Lipschitz domain and solutions to the heat equation on a time-varying domain. Our purpose is twofold: (a) to characterize the Radon–Nikodym property of the Banach space X in terms of the existence of nontangential limits of X-valued functions u in the corresponding Hardy space with index p≥1, (b) to identify the function of the boundary values of u in the Hardy space with index p>1 with an element in the space VXp of measures of p-bounded variation in the absence of the Radon–Nikodym property of X. This extends similar results already known on the unit disk of C and the semispace Rn×(0,∞).