Extensions of harmonic and analytic functions

Journal D Analyse Mathematique, 2006

For 0 < p ≤ ∞ and 0 < q ≤ ∞, the space of Hardy-Bloch type B(p, q) consists of those functions f which are analytic in the unit disk D such that (1 − r)Mp(r, f ) ∈ L q (dr/(1 − r)). We note that B(∞, ∞) coincides with the Bloch space B and that B ⊂ B(p, ∞), for all p. Also, the space B(p, p) is the Dirichlet space D p p−1 . We prove a number of results on decomposition of spaces with logarithmic weights which allow us to obtain sharp results about the mean growth of the B(p, q)-functions. In particular, we prove that if f is an analytic function in D and 2 ≤ p < ∞, then the condition Mp(r, f ) = O (1 − r) −1 ¡ , as r → 1, implies that Mp(r, f ) = O log 1 1−r 1/2 , as r → 1. This result is an improvement of the well known estimate of Clunie and MacGregor and Makarov about the integral means of Bloch functions, and it also improves the main result in a recent paper by Girela and Peláez. We also consider the question of characterizing the univalent functions in the spaces B(p, 2), 0 < p < ∞, and in some other related spaces and give some applications of our estimates to study the Carleson measures for the spaces B(p, 2) and D p p−1 . D |f (z)| p dA(z) < ∞ 2000 Mathematics Subject Classification. 30D45, 30D55.

Bulletin of The American Mathematical Society, 1989

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.

Proceedings of the American Mathematical Society, 2018

We study the spectrum M b (U) of the algebra of bounded type holomorphic functions on a complete Reinhardt domain in a symmetrically regular Banach space E as an analytic manifold over the bidual of the space. In the case that U is the unit ball of ℓ p , 1 < p < ∞, we prove that each connected component of M b (B ℓp) naturally identifies with a ball of a certain radius. We also provide estimates for this radius and in many natural cases we have the precise value. As a consequence, we obtain that for connected components different from that of evaluations, these radii are strictly smaller than one, and can be arbitrarily small. We also show that for other Banach sequence spaces, connected components do not necessarily identify with balls.

Pacific Journal of Mathematics, 1996

International Journal of Mathematics and Mathematical Sciences, 1990

The radial limits of the weighted derivative of an bounded analytic function is considered.

Mathematical Surveys and Monographs, 1980

As usual we let R be any discrete subring of the complex numbers. Throughout this chapter R will always contain the identity. DEFINITION 11.1. Let/be a complex valued function on a subset X of C. For each n in N set £"(/)= inf \\f-p\\ x deg/></i where the/?'s have arbitrary complex coefficients and £"<(/)= inf ||/-9H* deg q < n where the coefficients of the q's he in R. Clearly E Q {f) > E x (f) > , ES(f) > W) > , 0) and E"(f) < E< n (f), n e N. (2) In Part II we were concerned with characterizing those/for which E*(f)->0 as n-» oo. In Part III we will obtain information about the asymptotic behavior of the sequence {E*(f)} as n-> oo. From (2) we have a sort of lower bound on the rate of convergence of E*(f) to zero which we will not mention again. In the present chapter we will be concerned with the effect on the sequence {E*(f)} of the hypothesis that/be analytic. If/is analytic on an interval, then by a well-known theorem of S. N. Bernstein (Lorentz [66, 5.5]) there exists p < 1 such that E n (f) = 0(p n), i.e., there exists a constant C such that E n (f) < Cp n , all n in N. A similar result holds for E%(f). We start with the following. THEOREM 11.2. Let q be a monic polynomial in R[z], 0 < p, < p 2 , and z v. .. , z k the zeros of q. If f is analytic where \q{z)\ < p 2 and X = (z: \q(z)\ < pj}, then E£(f) = 0(p n) on X for some p < \ if and only if for every positive integer r there is a polynomial q r in R [z] satisfying tf\zj) = f (*\zj), \<j<k,0<s<r.

2009

Let a ∈ C n. Denote L a the bunch of complex lines containing a. It is easy to construct a real-analytic, and even polynomial, function f on the unit complex sphere ∂B n in C n such that f is the boundary value of no holomorphic function in the ball B n , but nevertheless, for any complex line L ∈ L a the restriction f | L∩∂B n continuously extends in L ∩ B n as a function, holomorphic in L ∩ B n. We prove that, however, two bunches of complex lines are sufficient for testing global holomorphic extendibility of real-analytic functions: if a and b are two distinct points in the closed unit ball and f ∈ C ω (∂B n) admits the one-dimensional holomorphic extension in any complex line L ∈ L a ∪ L b then f is the boundary value of a holomorphic function in the unit ball B n .

2010

The initial motivation for this paper is to discuss a more concrete approach to an approximation theorem of Axler and Shields, which says that the uniform algebra on the closed unit disc closure(D) generated by z and h --- where h is a nowhere-holomorphic harmonic function on D that is continuous up to the boundary --- equals the algebra of continuous functions on closure(D). The abstract tools used by Axler and Shields make harmonicity of h an essential condition for their result. We use the concepts of plurisubharmonicity and polynomial convexity to show that, in fact, the same conclusion is reached if h is replaced by h+R, where R is a non-harmonic perturbation whose Laplacian is "small" in a certain sense.

2003

Systems of analytic functions which are simultaneously orthogonal over each of two domains were apparently first studied in particular cases by Walsh and Szego¨, and in full generality by Bergman. In principle, these are very interesting objects, allowing application to analytic continuation that is not restricted (as Weierstrassian continuation via power series) either by circular geometry or considerations of locality. However, few explicit examples are known, and in general one does not know even gross qualitative features of such systems. The main contribution of the present paper is to prove qualitative results in a quite general situation.