Potential space estimates in local Hardy spaces for Green potentials in convex domains

Let Ω be a bounded co.nvex domain in Rn(n≥3) and G(x,y) be the Green function of the Laplace operator -△ on Ω. Let hrp(Ω) = {f ∈ D'(Ω) :(E)F∈hp(Rn), s.t. F|Ω = f}, by the atom characterization of Local Hardy spaces in a bounded Lipschitz domain, the bound of f→(△)2(Gf) for every f ∈ hrp(Ω) is obtained, where n/(n 1)＜p≤1.     

Coefficients of nonvanishing functions inH ∞

LesB denote the class of functions analytic in the unit disc ofC which satisfy 0<|f(z)|<1. It is proved that there exists a numberc<1 such that iff∈B and iff(z)=Σ _n=0 ^∞ a _n z ⁿ , then |a _n |<c forn>=1.     

On the dynamics of composite entire functions

Letf andg be nonlinear entire functions. The relations between the dynamics off⊗g andg⊗f are discussed. Denote byℐ (·) andF(·) the Julia and Fatou sets. It is proved that ifz∈C, thenz∈ℐ8464 (f⊗g) if and only ifg(z)∈ℐ8464 (g⊗f); ifU is a component ofF(f○g) andV is the component ofF(g○g) that containsg(U), thenU is wandering if and only ifV is wandering; ifU is periodic, then so isV and moreover,V is of the same type according to the classification of periodic components asU. These results are used to show that certain new classes of entire functions do not have wandering domains. The second author was supported by Max-Planck-Gessellschaft ZFDW, and by Tian Yuan Foundation, NSFC.     

Almost homomorphisms between unital <Emphasis Type="Italic">C</Emphasis>*-algebras: A fixed point approach

Let A , B be two unital C*-algebras. By using fixed pint methods, we prove that every almost unital almost linear mapping h : A → B which satisfies h(2 n uy) = h(2 n u)h(y) for all u ∈ U(A), all y ∈ A, and all n = 0, 1, 2, … , is a homomorphism. Also, we establish the generalized Hyers-Ulam-Rassias stability of *-homomorphisms on unital C*-algebras.     

Conjugation of injections by permutations

Let Ω be a countably infinite set, Inj(Ω) the monoid of all injective endomaps of Ω, and Sym(Ω) the group of all permutations of Ω. Also, let f,g,h∈Inj(Ω) be any three maps, each having at least one infinite cycle. (For instance, this holds if f,g,h∈Inj(Ω)∖Sym(Ω).) We show that there are permutations a,b∈Sym(Ω) such that h=afa ⁻¹ bgb ⁻¹ if and only if |Ω∖(Ω)f|+|Ω∖(Ω)g|=|Ω∖(Ω)h|. We also prove a generalization of this statement that holds for infinite sets Ω that are not necessarily countable.     

Carathéodory balls and norm balls of the domainH={(z1,z2)∈C2∶|z1|+|z2|<1}

LetH be the domain inC ² defined byH={Z=(z ₁,z ₂):║Z║₁=│z║₁│+│z║₂│<1}. LetC _H(z,w) be the Carathéodory distance ofH,z,w∈H. The Carathéodory ballB _C(z_C,α;H) with centerz _C,z_C∈H, and radius α, 0<α<1, is defined byB _c(z_C,α;H)={z∶C_H(z,z_C)<arc tanh α}. The norm ballB _N(z_N,r) with centerz _N,z_N∈H, and radiusr, 0<r<1-‖z _N‖₁, is defined byB _N(z_N,r)={z∶ ‖z−z_N‖₁<r}. Theorem:The only Carathéodory balls of H which are also norm balls are those with their center at the origin.     

Convolution properties of some classes of analytic functions

Let A denote the class of functions which are analytic in |z|<1 and normalized so that f(0)=0 and f′(0)=1, and let R(α, β)⊂A be the class of functions f such thatRe[f′(z)+αzf″(z)]>β,Re α>0, β<1. We determine conditions under which (i) f ∈ R(α₁, β₁), g ∈ R(α₂, β₂) implies that the convolution f×g of f and g is convex; (ii) f ∈ R(0, β₁), g ∈ R(0, β₂) implies that f×g is starlike; (iii) f≠A such that f′(z)[f(z)/z]^μ-1 ≺ 1 + λz, μ>0, 0<λ<1, is starlike, and (iv) f≠A such that f′(z)+αzf″(z) ≺ 1 + λz, α>0, δ>0, is convex or starlike. Bibliography: 16 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 138–154.     

Characterizations of Hardy-Orlicz and Bergman-Orlicz spaces

Let and τ denote the invariant gradient and invariant measure on the unit ball B of ℂⁿ, respectively. Assume that f is a holomorphic function on B and ϕ ∈ C²(ℝ) is a nonnegative, nondecreasing, convex function. Then f belongs to the Hardy-Orlicz space H _ϕ(B>) if and only if

Families of meromorphic functions avoiding continuous functions

Leth ₁,h ₂ andh ₃ be continuous functions from the unit disk D into the Riemann sphereC such thath _i(z) ≠ h_j(z) (i ≠ j) for eachz∈D. We prove that the setF of all functionsf meromorphic on D such thatf(z)≠h _j (z) for allz ∈ D andj=1,2,3 is a normal family. The result and the method of the proof extend to quasimeromorphic functions in higher dimensions as well. The second author was supported by a Heisenberg fellowship of the DFG. The fourth author was partially supported by the Marsden Fund, New Zealand. This research was completed while the authors were attending a conference at Mathematisches Forschungsinstitut Oberwolfach in Germany. The authors would like to express their sincere thanks to the Institute for providing a stimulating atmosphere and for its kind hospitality.     

On the height of the Kronecker product ofS n characters

Given any two heightsh ₁,h ₂, we can choose wide enough partitionsν, μ ∈ Par(n) such thath(ν)=h ₁,h(μ)=h ₂ andh(x _ν⊗xμ)=h ₁sdh ₂.     

Cyclic elements of the shift operator in weighted anisotropic spaces of holomorphic functions in the polydisk

Let ϕ(r) = (ϕ₁(r₁), …, ϕ_n(r_n)) be a vector-valued function on R ₊ ⁿ . A necessary and sufficient condition is obtained under which any function f ∈, H^∞ (D ⁿ ), f(z) ≠ 0, z ∈, D ⁿ , is cyclic in the corresponding weighted space L^p(ϕ), where D ⁿ is the unit polydisk in C ⁿ. Bibliography: 13 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 327, 2005, pp. 226–234.     

A Property About Minimum Edge- and Minimum Clique-Cover of a Graph

 Let G be a graph with n vertices, and denote as γ(G) (as θ(G)) the cardinality of a minimum edge cover (of a minimum clique cover) of G. Let E (let C) be the edge-vertex (the clique-vertex) incidence matrix of G; write then P(E)={x∈ℜ ⁿ :Ex≤1,x≥0}, P(C)={x∈ℜ ⁿ :Cx≤1,x≥0}, α _E (G)=max{1 ^T x subject to x∈P(E)}, and α _C (G)= max{1 ^T x subject to x∈P(C)}. In this paper we prove that if α _E (G)=α _C (G), then γ(G)=θ(G). Received: May 20, 1998?Final version received: April 12, 1999     

Quasiconformal and harmonic mappings between Jordan domains

Let Ω and Ω₁ be Jordan domains, let μ ∈ (0, 1], and let be a harmonic homeomorphism. The object of the paper is to prove the following results: (a) If f is q.c. and ∂Ω, ∂Ω₁ ∈ C ^1,μ , then f is Lipschitz; (b) if f is q.c., ∂Ω, ∂Ω₁ ∈ C ^1,μ and Ω₁ is convex, then f is bi-Lipschitz; and (c) if Ω is the unit disk, Ω₁ is convex, and ∂Ω₁ ∈ C ^1,μ , then f is quasiconformal if and only if its boundary function is bi-Lipschitz and the Hilbert transform of its derivative is in L ^∞. These extend the results of Pavlović (Ann. Acad. Sci. Fenn. 27:365–372, 2002).      

Factorization of Hardy spaces and Hankel operators on convex domains in ℂ n

In this article we study the (small) Hankel operator h_b on the Hardy and Bergman spaces on a smoothly bounded convex domain of finite type in ℂⁿ. We completely characterize the Hankel operators h_b that are bounded, compact, and belong to the Schatten ideal S_p, for 0 < p < ∞. In particular, if h_b denotes the Hankel operator on the Hardy space H² (Ω), we prove that h_b is bounded if and only if b ∈ BMOA, compact if and only if b ∈ VMOA, and in the Schatten class if and only if b ∈e B_p, 0 < p < ∞. This last result extends the analog theorem in the case of the unit disc of Peller [19] and Semmes [21]. In order to characterize the bounded Hankel operators, we prove a factorization theorem for functions in H¹ (Ω), a result that is of independent interest.     

WeightedL p closure theorems for spaces of entire functions

For a fixed weight Δ(dx) onR ¹ and a linear space ℋ ⊆L ^p(Δ) of entire functions that is closed under difference quotientsh(·)→(z−·) ⁻¹[h(z)−h(·)], theL ^p(Δ) closure of ℋ is studied and characterized in terms of the normsL(z), (z∈C ¹ of the evaluation functionalsh→h(z),h∈ℋ. Partially supported by DA-ARO-31-124-71-6182 and NSF GP-43011.     

A note on regularity for a class of quasilinear elliptic equations

The following regularity of weak solutions of a class of elliptic equations of the form are investigated.     

Closures of finitely generated ideals in Hardy spaces

LetH ^∞ be the algebra of bounded analytic functions in the unit diskD. LetI=I(f ₁,...,f _N) be the ideal generated byf ₁,...,f _N∈H ^∞ andJ=J(f ₁,...,f _N) the ideal of the functionsf∈H ^∞ for which there exists a constantC=C(f) such that |f(z)|≤C(|f ₁ (z)|+...;+|f _N (z)|),z∈D. It is clear that , but an example due to J. Bourgain shows thatJ is not, in general, in the norm closure ofI. Our first result asserts thatJ is included in the norm closure ofI ifI contains a Carleson-Newman Blaschke product, or equivalently, if there existss>0 such that

Renorming in the space of Bochner integrable functionsL 1(μ,X)

A sufficient condition is given when a subspaceL⊂L ₁(μ,X) of the space of Bochner integrable function, defined on a finite and positive measure space (S, Φ, μ) with values in a Banach spaceX, is locally uniformly convex renormable in terms of the integrable evaluations {∫ _A fdμ;f∈L}. This shows the lifting property thatL ₁(μ,X) is renormable if and only ifX is, and indicates a large class of renormable subspaces even ifX does not admit and equivalent locally uniformly convex norm.     

Punishing Factors for Finitely Connected Domains

Let Ω and Π be two finitely connected hyperbolic domains in the complex plane \Bbb C{\Bbb C} and let R(z, Ω) denote the hyperbolic radius of Ω at z and R(w, Π) the hyperbolic radius of Π at w. We consider functions f that are analytic in Ω and such that all values f(z) lie in the domain Π. This set of analytic functions is denoted by A(Ω, Π). We prove among other things that the quantities C_n(W,P) := sup_{f ? A(W,P)}sup_{z ? W}\frac|f⁽ⁿ⁾(z)| R(f(z),P)n! (R(z,W))ⁿC_n(\Omega,\Pi)\,:=\,\sup_{f\in A(\Omega,\Pi)}\sup_{z\in \Omega}\frac{\vert f^{(n)}(z)\vert\,R(f(z),\Pi)}{n!\,(R(z,\Omega))^n} are finite for all n ? \Bbb N{n \in {\Bbb N}} if and only if ∂Ω and ∂Π do not contain isolated points.