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

Composition operators on the Lipschitz space in polydiscs

Let Un be the unit polydisc of Cn and φ= (φ1,...,φn? a holomorphic self-map of Un. Let 0≤α< 1. This paper shows that the composition operator Cφ, is bounded on the Lipschitz space Lipa(Un) if and only if there exists M > 0 such thatfor z∈Un. Moreover Cφ is compact on Lipa(Un) if and only if Cφ is bounded on Lipa(Un) and for every ε > 0, there exists a δ > 0 such that whenever dist(φ(z),σUn) <δ     

Entire functions, analytic continuation, and the fractional parts of a linear function

The main result of the paper is as follows.Theorem. Suppose that G(z) is an entire function satisfying the following conditions: 1) the Taylor coefficients of the function G(z) are nonnegative: 2) for some fixed C>0 and A>0 and for |z|>R₀, the following inequality holds:

Bloch-Bergman Pullbacks with Logarithmic Weights

We characterize the composition operators mapping Blochs boundedly into the weighted Bergman spaces of logarithmic weight. For 0 < p < ∞, 1 < α < ∞, let A^{p, log α} denote the space of holomorphic functions F in the unit disc D for which

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.     

Estimates in the corona theorem and ideals ofH ∞ : A problem of T. Wolff

The main result of the paper is that there exist functionsf ₁,f ₂,f inH ^∞ satisfying the “corona condition”

Proof of a hypercontractive estimate via entropy

Consider the probability spaceW={−1, 1} ⁿ with the uniform (=product) measure. Letf: W →R be a function. Letf=Σf _IX_I be its unique expression as a multilinear polynomial whereX _I=Π _i∈I x _i. For 1≤m≤n let =Σ_|I|=m f _IX_I. LetT _ɛ (f)=Σf _Iɛ^|I| X _I where 0<ɛ<1 is a constant. A hypercontractive inequality, proven by Bonami and independently by Beckner, states that

Regularity results for a class of obstacle problems

We prove some optimal regularity results for minimizers of the integral functional ∫ f(x, u, Du) dx belonging to the class K ≔ {u ∈ W ^1,p (Ω): u ⩾ ψ, where ψ is a fixed function, under standard growth conditions of p-type, i.e.

Null-sets ofH-solutions

We considerC ²-solutionsf=u+iv+jw of the system

On zeros of functions of given proximate formal order analytic in a half-plane

We describe sequences of zeros of functionsf≢0 that are analytic in the half-plane ℂ₊={z:Rez> and satisfy the condition

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

Univalence and starlikeness of nonlinear integral transform of certain class of analytic functions

Let U(λ, μ) denote the class of all normalized analytic functions f in the unit disk |z| < 1 satisfying the condition

On the decay rate of (p, A)-lacunary series

A power series with radius of convergence equal 1 is called a (p,A)-lacunary one if n_k ≥ Ak^p, A > 0, 1 < p < ∞. It is proved that if 1 < p < 2 and f(x) is a (p,A)-lacunary series that satisfies the condition

Approximation of several dimensional functions by trigonometric polynomials

Let f(x, y) be a periodic function defined on the region D

On the riemann zeta-function and the divisor problem II

Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of . If E ^*(t)=E(t)-2πΔ^*(t/2π) with , then we obtain

Boundary functions in L2 H $$(\mathbb{B}^n )$$

We solve the Dirichlet problem for line integrals of holomorphic functions in the unit ball For a function u which is lower semi-continuous on we give necessary and sufficient conditions in order that there exists a holomorphic function f ∈ such that

On a theorem of Baernstein

In the paper [B2] Baernstein constructs a simply connected domain Θ in the plane for which the conformal mappingf of Θ into the unit disc Δ satisfies

Compact composition operators on the Bloch space in polydiscs

Let Uⁿ be the unit polydisc of ℂⁿ and φ=(φ₁, ⋯, φ _n ) a holomorphic self-map of Uⁿ. As the main result of the paper, it shows that the composition operator C is compact on the Bloch space β(Uⁿ) if and only if for every ε > 0, there exists a δ > 0, such that

On the growth of universal functions

In this paper, we study the growth of universal functions (Taylor series) on the unit discD. We describe a class of growth types prohibited for such a function. By investigating the relation between growth and value distribution, we prove that every universal function assumes every complex value, with at most one exception, on sequences inD that approach ϖD rather slowly. We use this to get a large class of equations, including polynomials with Nevanlinna coefficients and equations of iterates, that no universal function satisfies. Finally, we produce a universal function whose growth is bounded by