Hardy spaces of close-to-convex functions and their derivatives

Rocky Mountain Journal of Mathematics, 2005

Complex Analysis and Operator Theory, 2014

In this paper we mainly discuss three things. First, there is no canonical norm on the space H p u (D). Second, we improve the weak-* convergence of the measures µu,r. Third, the dilations ft of the function f ∈ H p u (D) converge to f in H p u-norm and hence the polynomials are dense in H p u (D).

2018

In this paper, we start by proving that the function which is holomorphic in the open unit disc centred at the origin, is an element of a Hardy space if and only if Here we give a new proof for a known result. Moreover, the present work provides two different new proofs for one of the implications mentioned above. One proves that the same function is an element of a Bergman space if and only if This is the first completely new result of this work. From these theorems we deduce the behavior of the function in the half – open disc Although the assertions claimed above refer to complex analytic functions, and the involved spaces are function spaces of analytic complex functions, the proofs from below are based on results and methods of real analysis.

Journal of Functional Analysis, 1989

Let BP= {f:))fj) =sup,,, (1/2T)~~,lflP)*'P< co), 1 <p< co. Then BP is the dual of a function algebra Aq on R (Beurling). In this paper, we study the harmonic extensions offin BP and in A", and the corresponding Hardy spaces H,,, HA*. It is shown that a parallel theory for L", L' and BMO, H' can be developed for the above pairs. In particular we prove that for 1 < q < 2, (HAM)* is isomorphic to the Banach space where mTf= (1/2T) I',! We also prove Burkholder, Gundy, and Silverstein's maximal function characterization for the new Hardy space HA', 1 < 4 < 2. 0 1989 Academic Press, Inc.

Journal D Analyse Mathematique, 1997

This paper is concerned with several approximation problems in the weighted Hardy spacesH p(Ω) of analytic functions in the open unit disc D of the complex plane ℂ. We prove that ifX is a relatively closed subset of D, the class of uniform limits onX of functions inH p(Ω) coincides, moduloH p(Ω), with the space of uniformly continuous functions on a certain hull ofX which are holomorphic on its interior. We also solve the simultaneous approximation problems of describing Farrell and Mergelyan sets forH p(Ω), giving geometric characterizations for them. By replacing approximating polynomials by polynomial multipliers of outer functions, our results lead to characterizations of the same sets with respect to cyclic vectors in the classical Hardy spacesH p(D), 1 ⪯p < ∞.

Complex Variables and Elliptic Equations, 2019

Analytic functions defined on a tube domain T C ⊂ C n and taking values in a Banach space X which are known to have X-valued distributional boundary values are shown to be in the Hardy space H p (T C , X) if the boundary value is in the vector valued Lebesgue space L p (R n , X), where 1 ≤ p ≤ ∞ and C is a regular open convex cone. Poisson integral transform representations of elements of H p (T C , X) are also obtained for certain classes of Banach spaces, including reflexive Banach spaces.

We present some inequalities for the Taylor coefficients of a Hardy-Orlicz function, which in the case of the standard Hardy spaces reduce to the inequalities of Hardy and Littlewood. The proofs are new and elementary even in the case of the classical Hardy spaces. We also prove a Marcinkiewicz type theorem that extends a theorem of Kislyakov and Xu. At the end, we prove a Hardy-Prawitz criterion for membership of a univalent function in a Hardy-Orlicz space.

Harmonic Analysis, Partial Differential Equations, Complex Analysis, Banach Spaces, and Operator Theory (Volume 1), 2016

We prove various representations and density results for Hardy spaces of holomorphic functions for two classes of bounded domains in C n , whose boundaries satisfy minimal regularity conditions (namely the classes C 2 and C 1,1 respectively) together with naturally occurring notions of convexity.

Lecture Notes in Mathematics, 1989

Previous chapters were preliminary in the sense that they covered some of the basic results leading to the theory of the weighted Hardy spaces. Another important property, namely, the fact that the "norm" of a distribution in these spaces can be computed in various equivalent ways, is the content of this chapter.