[Paper Review] Radial growth of functions from the Korenblum space

Alexander Borichev, Yurii Lyubarskii+2 more|arXiv (Cornell University)|Mar 30, 2009

Holomorphic and Operator Theory6 references3 citations

TL;DR

This paper investigates the radial growth and decay of analytic and harmonic functions in the unit disk that are majorized by the Korenblum-type majorant $ v(r) = \\-\log(1-r)^{-1} $. Using Hausdorff measure estimates, it shows that extremal radial growth or decay can only occur on exceptional sets of zero Lebesgue measure, with precise quantitative bounds on the size of these sets in terms of the scale $ h_\alpha(t) = t|\log t|^\alpha $. The key contribution is sharp estimates for the exceptional sets using Hausdorff measures.

ABSTRACT

We study radial behavior of analytic and harmonic functions, which admit a certain majorant in the unit disk. We prove that extremal growth or decay may occur only along small sets of radii and give precise estimates of these exceptional sets.

Motivation & Objective

• To understand the radial behavior of analytic and harmonic functions in the unit disk that are majorized by $ v(r) = \log\frac{1}{1-r} $, a model for the Korenblum space $ A^{-\infty} $.
• To determine how fast such functions can grow or decay along radii, particularly whether extremal growth/decay can occur on large sets of radii.
• To quantify the size of the exceptional sets of radii where extremal radial growth or decay may occur, using Hausdorff measures with gauge functions $ h_\alpha(t) = t|\log t|^\alpha $.
• To show that for functions in the Korenblum class, such extremal behavior is restricted to sets of zero Lebesgue measure, with precise measure-theoretic bounds.
• To demonstrate sharpness of the estimates via explicit constructions of functions and Cantor-type sets with controlled Hausdorff measure.

Proposed method

• Define the sets $ D_+(f) $ and $ D_-(f) $ as the sets of angles $ \theta \in [0,2\pi) $ where $ \log|f(re^{i\theta})| $ grows or decays faster than $ |\log(1-r)| $, respectively.
• Use the Poisson integral representation to relate harmonic functions to boundary measures, and derive estimates for non-tangential growth from radial growth.
• Apply harmonic measure inequalities and maximal function estimates to control the size of exceptional sets where radial growth or decay exceeds the majorant.
• Construct Cantor-type sets $ C $ with prescribed Hausdorff measure $ H_\lambda(C) $ using nested intervals and dyadic decompositions to test sharpness of estimates.
• Use the gauge function $ \lambda(t) = t|\log t|^\alpha $ to define Hausdorff measures and derive lower bounds on $ H_\lambda(D_\pm(f)) $ via covering arguments and recursive interval counting.
• Construct a sequence of positive harmonic functions $ v^{(n)} $ with $ v^{(n)} \in \mathcal{K} $, such that $ H_\lambda(E_+(v^{(n)})) \to \infty $, proving the sharpness of the bound.

Experimental results

Research questions

• RQ1 What is the maximal possible size of the set of radii along which a function in the Korenblum space exhibits radial growth faster than $ \log\frac{1}{1-r} $?
• RQ2 Can a harmonic function in the Korenblum space decay to $ -\infty $ along a large set of radii faster than $ -\log\frac{1}{1-r} $, and if so, how large can such a set be?
• RQ3 How do the exceptional sets for radial growth and decay compare in size, and can they be quantified using Hausdorff measures with gauge functions $ h_\alpha(t) = t|\log t|^\alpha $?
• RQ4 Is the estimate on the Hausdorff measure of the exceptional set for radial growth sharp, and can it be saturated by explicit constructions?
• RQ5 Does the behavior change significantly when considering positive harmonic functions satisfying the majorant condition?

Key findings

• For any function $ f \in A^{-\infty} $, the set $ D_+(f) $ of radii along which $ \log|f(re^{i\theta})| $ grows faster than $ |\log(1-r)| $ has Hausdorff measure zero with respect to any gauge function $ \lambda(t) = o(t|\log t|^\alpha) $ as $ t \to 0 $, for every $ \alpha > 0 $.
• The exceptional set $ D_+(f) $ has positive $ H_\lambda $-measure for $ \lambda(t) = t|\log t|^\alpha $ if $ \alpha \geq 1 $, and the bound is sharp: there exist functions for which $ H_\lambda(D_+(f)) > 0 $ for such $ \lambda $.
• For harmonic functions $ u \in \mathcal{K} $, the set $ D_-(u) $ of radii along which $ u(re^{i\theta}) \to -\infty $ faster than $ \log\frac{1}{1-r} $ also satisfies the same Hausdorff measure estimates as $ D_+(f) $, indicating symmetry in the exceptional set size.
• The paper constructs a function $ u \in \mathcal{K} $ such that $ H_\lambda(E_+(u)) = \infty $ for $ \lambda(t) = t|\log t| $, showing that the bound is sharp and cannot be improved.
• For positive harmonic functions satisfying $ u(z) \leq \log\frac{1}{1-|z|} $, the exceptional set $ E_+(u) $ for radial growth has full Hausdorff measure $ H_\lambda(E_+(u)) = \infty $ for $ \lambda(t) = t|\log t| $, indicating a different behavior compared to general harmonic functions.
• The construction of Cantor-type sets $ C^{(n)} $ with $ H_\lambda(C^{(n)}) \to \infty $ as $ n \to \infty $, while remaining in the exceptional sets of positive harmonic functions, confirms the sharpness of the measure-theoretic estimates.