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Bloch functions and asymptotic tail variance
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). Saint Petersburg State University,Russian Federation.ORCID iD: 0000-0002-4971-7147
2017 (English)In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 313, 947-990 p.Article in journal (Refereed) Published
Abstract [en]

Let P denote the Bergman projection on the unit disk D, P mu(z) := integral(D) mu(w)/(1-z (w) over bar)(2) dA(w), z is an element of D, where dA is normalized area measure. We prove that if vertical bar mu(z)vertical bar <= 1 on D, then the integral I(mu()a,r) := integral(2 pi)(0) exp {a r(4)vertical bar P mu(re(i theta))vertical bar(2)/log 1/1-r(2)}d theta/2 pi, 0 < r < 1, has the bound I-mu(a,r) <= C(a) := 10(1-a)(-3/2) for 0 < a < 1, irrespective of the choice of the function mu. Moreover, for a > 1, no such uniform bound is possible. We interpret the theorem in terms the asymptotic tail variance of such a Bergman projection P-mu (by the way, the asymptotic tail variance induces a seminorm on the Bloch space). This improves upon earlier work of Makarov, which covers the range 0 < a < pi(2)/64 = 0.1542.... We then apply the theorem to obtain an estimate of the universal integral means spectrum for conformal mappings with a k-quasiconformal extension, for 0 < k < 1. The estimate reads, for t is an element of C and 0 < k < 1, B(k,t) <= {1/4 k(2)vertical bar t vertical bar(2)(1 + 7k)(2), for vertical bar t vertical bar <= 2/k(1 + 7k)(2), k vertical bar t vertical bar - 1/(1 + 7k)(2), for vertical bar t vertical bar >= 2/k(1 + 7k)(2), which should be compared with the conjecture by Prause and Smirnov to the effect that for real t with vertical bar t vertical bar <= 2/k, we should have B(k, t) = 1/4k(2)t(2).

Place, publisher, year, edition, pages
Academic Press, 2017. Vol. 313, 947-990 p.
Keyword [en]
Asymptotic variance, Asymptotic tail variance, Bloch function, Bergman projection, Quasiconformal, Holomorphic motion
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-209286DOI: 10.1016/j.aim.2017.04.016ISI: 000402222700024Scopus ID: 2-s2.0-85018989478OAI: oai:DiVA.org:kth-209286DiVA: diva2:1111530
Note

QC 20170619

Available from: 2017-06-19 Created: 2017-06-19 Last updated: 2017-06-19Bibliographically approved

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