Extreme points, support points and the Loewner variation in several complex variables

In this paper we consider extreme points and support points for compact subclasses of normalized biholomorphic mappings of the Euclidean unit ball Bn in Cn. We consider the class S0(Bn) of biholomorphic mappings on Bn which have parametric representation, i.e., they are the initial elements f (·, 0) of a Loewner chain f (z, t) = etz + ··· such that {e-tf (·, t)}t 0 is a normal family on Bn. We show that if f (·, 0) is an extreme point (respectively a support point) of S0(Bn), then e-tf (·, t) is an extreme point of S0(Bn) for t 0 (respectively a support point of S0(Bn) for t ∈[0, t0] and some t0 > 0). This is a generalization to the n-dimensional case of work due to Pell. Also, we prove analogous results for mappings which belong to S0(Bn) and which are bounded in the norm by a fixed constant. We relate the study of this class to reachable sets in control theory generalizing work of Roth. Finally we consider extreme points and support points for biholomorphic mappings of Bn generated by using extension operators that preserve Loewner chains.     

A class of Loewner chain preserving extension operators

We consider operators that extend locally univalent mappings of the unit disk Δ in C to locally biholomorphic mappings of the Euclidean unit ball B of Cn. For such an operator Φ, we seek conditions under which etΦ(e−tf(⋅,t)), t?0, is a Loewner chain on B whenever f(⋅,t), t?0, is a Loewner chain on Δ. We primarily study operators of the form , , where β∈[0,1/2] and is holomorphic, finding that, for ΦG,β to preserve Loewner chains, the maximum degree of terms appearing in the expansion of G is a function of β. Further applications involving Bloch mappings and radius of starlikeness are given, as are elementary results concerning extreme points and support points.     

Growth theorems and coefficient bounds for univalent holomorphic mappings which have parametric representation

Let B be the unit ball in Cn with respect to an arbitrary norm and let f(z,t) be a g-Loewner chain such that e−tf(z,t)−z has a zero of order k+1 at z=0. In this paper, we obtain growth and covering theorems for . Moreover, we consider coefficient bounds and examples of mappings in .     

Polynomially bounded solutions to the Loewner differential equation in several complex variables

We determine the form of polynomially bounded solutions to the Loewner differential equation that is satisfied by univalent subordination chains of the form f(z,t)=etAz+?, where A∈L(Cn,Cn) has the property m(A)>0. Here m(A)=min{R〈A(z),z〉:‖z‖=1}. We also give sufficient conditions for g(z,t)=L(f(z,t)) to be polynomially bounded, where f(z,t) is an A-normalized polynomially bounded Loewner chain solution to the Loewner differential equation.     

On normal families and differential polynomials for meromorphic functions

We consider the normality criterion for a families F meromorphic in the unit disc Δ, and show that if there exist functions a(z) holomorphic in Δ, a(z)≠1, for each z∈Δ, such that there not only exists a positive number ε₀ such that |an(a(z)−1)−1|?ε₀ for arbitrary sequence of integers an(n∈N) and for any z∈Δ, but also exists a positive number B>0 such that for every f(z)∈F, B|f^′(z)|?|f(z)| whenever f(z)f^″(z)−a(z)(f^′2(z))=0 in Δ. Then is normal in Δ.     

Resonances in Loewner equations

We prove that given a Herglotz vector field on the unit ball of Cn of the form H(z,t)=(a₁z₁,…,anzn)+O(²|z|) with for all j, its evolution family admits an associated Loewner chain, which is normal if no real resonances occur. Hence the Loewner–Kufarev PDE admits a solution defined for all positive times.     

Solution of a Loewner chain equation in several complex variables

We find a solution to the Loewner chain equation in the case when the infinitesimal generator satisfies h(0,t)=0, Dh(0,t)=A for any A∈L(Cn,Cn) with m(A)>0. We also study the related classes of spirallike mappings, mappings with parametric representation and asymptotically spirallike mappings.     

Biholomorphic mappings on bounded starlike circular domains

Let Ω⊂Cn be a bounded starlike circular domain with 0∈Ω. In this paper, we introduce a class of holomorphic mappings Mg on Ω. Let f(z) be a normalized locally biholomorphic mapping on Ω such that and z=0 is the zero of order k+1 of f(z)−z. We obtain a sharp growth theorem and sharp coefficient bounds for f(z). As applications, sharp distortion theorems for a subclass of starlike mappings are obtained. These results unify and generalize many known results.     

Integral representations for a class of operators

Let X be a completely regular Hausdorff space, E Hausdorff a quasi-complete locally convex space and Cb(X,E) all E-valued bounded continuous functions on X with strict topologies βt, , . We prove that a linear continuous mapping T:Cb(X,E)→E arises from a scalar measure μ∈(Cb^′(X),βz)(z=t,∞,τ) if and only if g(T(f))=0 whenever g○f=0 for any f∈Cb(X,E), g∈E^′.     

Carleson embeddings for Sobolev spaces via heat equation

Let u(t,x) be the solution of the heat equation (∂_t-Δ_x)u(t,x)=0 on subject to u(0,x)=f(x) on Rⁿ. The main goal of this paper is to characterize such a nonnegative measure μ on that f(x)?u(t²,x) induces a bounded embedding from the Sobolev space , p∈[1,n) into the Lebesgue space , q∈(0,∞).     

The generalized Roper-Suffridge extension operator

Let f(z) be a normalized convex (starlike) function on the unit disc D. Let , where z=(z₁,z₂,…,z_n), z₁∈D, , p_i?1, i=2,…,n, are real numbers. In this note, we prove that Φ(f)(z)=(f(z₁),f′(z₁)^1/p₂z₂,…,f′(z₁)^1/p_nz_n) is a normalized convex (starlike) mapping on Ω, where we choose the power function such that (f′(z₁))^1/p_i|_z1=0=1, i=2,…,n. Some other related results are proved.     

Region of variability of two subclasses of univalent functions

Let F₁ (F₂ respectively) denote the class of analytic functions f in the unit disk |z|<1 with f(0)=0=f^′(0)−1 satisfying the condition RePf(z)<3/2 (RePf(z)>−1/2 respectively) in |z|<1, where Pf(z)=1+zf^″(z)/f^′(z). For any fixed z₀ in the unit disk and λ∈[0,1), we shall determine the region of variability for logf^′(z₀) when f ranges over the class and , respectively.     

Spectral conditions for admissibility of evolution equations in Hilbert space

We study properties of solutions of the evolution equation , where B is a closable operator on the space AP(R,H) of almost periodic functions with values in a Hilbert space H such that B commutes with translations. The operator B generates a family of closed operators on H such that (whenever eiλtx∈D(B)). For a closed subset Λ⊂R, we prove that the following properties (i) and (ii) are equivalent: (i) for every function f∈AP(R,H) such that σ(f)⊆Λ, there exists a unique mild solution u∈AP(R,H) of Eq. (∗) such that σ(u)⊆Λ; (ii) is invertible for all λ∈Λ and .     

A bilinear version of Orlicz-Pettis theorem

Given three Banach spaces X, Y and Z and a bounded bilinear map , a sequence x=n(xn)⊆X is called B-absolutely summable if is finite for any y∈Y. Connections of this space with are presented. A sequence x=n(xn)⊆X is called B-unconditionally summable if is finite for any y∈Y and z^∗∈Z^∗ and for any M⊆N there exists xM∈X for which _∑n∈M〈B(xn,y),z^∗〉=〈B(xM,y),z^∗〉 for all y∈Y and z^∗∈Z^∗. A bilinear version of Orlicz-Pettis theorem is given in this setting and some applications are presented.     

Selections of bounded variation under the excess restrictions

Let X be a metric space with metric d, c(X) denote the family of all nonempty compact subsets of X and, given F,G∈c(X), let e(F,G)=supx∈Finfy∈Gd(x,y) be the Hausdorff excess of F over G. The excess variation of a multifunction , which generalizes the ordinary variation V of single-valued functions, is defined by where the supremum is taken over all partitions of the interval [a,b]. The main result of the paper is the following selection theorem: If,V₊(F,[a,b])<∞,t₀∈[a,b]andx₀∈F(t₀), then there exists a single-valued functionof bounded variation such thatf(t)∈F(t)for allt∈[a,b],f(t₀)=x₀,V(f,[a,t₀))?V₊(F,[a,t₀))andV(f,[t₀,b])?V₊(F,[t₀,b]). We exhibit examples showing that the conclusions in this theorem are sharp, and that it produces new selections of bounded variation as compared with [V.V. Chistyakov, Selections of bounded variation, J. Appl. Anal. 10 (1) (2004) 1-82]. In contrast to this, a multifunction F satisfying e(F(s),F(t))?C(t−s) for some constant C?0 and all s,t∈[a,b] with s?t (Lipschitz continuity with respect to e(⋅,⋅)) admits a Lipschitz selection with a Lipschitz constant not exceeding C if t₀=a and may have only discontinuous selections of bounded variation if a<t₀?b. The same situation holds for continuous selections of when it is excess continuous in the sense that e(F(s),F(t))→0 as s→t−0 for all t∈(a,b] and e(F(t),F(s))→0 as s→t+0 for all t∈[a,b) simultaneously.     

Viscosity approximation methods for nonexpansive nonself-mappings

Let E a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E^∗, and K be a closed convex subset of E which is also a sunny nonexpansive retract of E, and be nonexpansive mappings satisfying the weakly inward condition and F(T)≠∅, and be a fixed contractive mapping. The implicit iterative sequence {xt} is defined by for t∈(0,1)

xt=P(tf(xt)+(1−t)Txt).     

Weak subordination for convex univalent harmonic functions

For two complex-valued harmonic functions f and F defined in the open unit disk Δ with f(0)=F(0)=0, we say f is weakly subordinate to F if f(Δ)⊂F(Δ). Furthermore, if we let E be a possibly infinite interval, a function with f(⋅,t) harmonic in Δ and f(0,t)=0 for each t∈E is said to be a weak subordination chain if f(Δ,t₁)⊂f(Δ,t₂) whenever t₁,t₂∈E and t₁<t₂. In this paper, we construct a weak subordination chain of convex univalent harmonic functions using a harmonic de la Vallée Poussin mean and a modified form of Pommerenke's criterion for a subordination chain of analytic functions.     

Structure of boundary blow-up solutions for quasi-linear elliptic problems II: small and intermediate solutions

The structure of positive boundary blow-up solutions to quasi-linear elliptic problems of the form −Δ_pu=λf(u), u=∞ on ∂Ω, 1<p<∞, is studied in a bounded smooth domain , for a class of nonlinearities f∈C¹((0,∞)?{z₂})∩C⁰[0,∞) satisfying f(0)=f(z₁)=f(z₂)=0 with 0<z₁<z₂, f<0 in (0,z₁)∪(z₂,∞), f>0 in (z₁,z₂). Large, small and intermediate solutions are obtained for λ sufficiently large. It is known from Part I (see Structure of boundary blow-up solutions for quasilinear elliptic problems, part (I): large and small solutions, preprint), that the large solution is the unique large solution to the problem. We will see that the small solution is also the unique small solution to the problem while there are infinitely many intermediate solutions. Our results are new even for the case p=2.