Nearly-optimal Estimates for the Stability Problem in Hardy Spaces Dang Duc Trong and Tuyen Trung Truong
نویسنده
چکیده
We continue the work of [14]. Let E be a non-Blaschke subset of the unit disc D of the complex plane C. Fixed 1 ≤ p ≤ ∞, let Hp(D) be the Hardy space of holomorphic functions in the disk whose boundary value function is in Lp(∂D). Fixed 0 < R < 1. For ǫ > 0 define Cp(ε,R) = sup{ sup |z|≤R |g(z)| : g ∈ H, ‖g‖p ≤ 1, |g(ζ)| ≤ ε ∀ζ ∈ E}. In this paper we find upper and lower bounds for Cp(ǫ, R) when ǫ 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 D and sets contained in a finite union of Stolz angles.
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