Nearly-optimal estimates for the stability problem in Hardy spaces

2008/11/29 by Trong, Dang Duc, Truong, Tuyen Trung
#30D15 #31A15 #44A10 #65F22 #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.0812.0075

Abstract

We continue the work of \citeTLNT. Let E be a non-Blaschke subset of the unit disc \mathbbD of the complex plane ℂ. Fixed 1≤ p≤ ∞, let H^p(\mathbbD) be the Hardy space of holomorphic functions in the disk whose boundary value function is in L^p(∂ \mathbbD). Fixed 00 define C_p(ε, R) = sup \sup_{|z| ≤ R}|g(z)|: g∈ H^p, ‖g‖_p≤ 1, |g(ζ)| ≤ ε ∀ ζ∈ E\. In this paper we find upper and lower bounds for C_p(ε, R) when ε is small for any non-Blaschke set E. The bounds are nearly-optimal for many such sets E, including sets contained in a compact subset of \mathbbD and sets contained in a finite union of Stolz angles.

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