Derivatives of Blaschke products whose zeros lie in a Stolz domain and weighted Bergman spaces

Reijonen, Atte

doi:10.1090/proc/13791

Derivatives of Blaschke products whose zeros lie in a Stolz domain and weighted Bergman spaces
HTML articles powered by AMS MathViewer

by Atte Reijonen PDF

Proc. Amer. Math. Soc. 146 (2018), 1173-1180 Request permission

Abstract:

For a Blaschke product $B$ whose zeros lie in a Stolz domain, we find a condition regarding $\omega$ which guarantees that $B’$ belongs to the Bergman space $A^p_\omega$. In addition, the sharpness of this condition is considered.

References

Patrick Ahern, The Poisson integral of a singular measure, Canad. J. Math. 35 (1983), no. 4, 735–749. MR 723040, DOI 10.4153/CJM-1983-042-0
P. R. Ahern and D. N. Clark, On inner functions with $H^{p}$-derivative, Michigan Math. J. 21 (1974), 115–127. MR 344479
Peter Colwell, Blaschke products, University of Michigan Press, Ann Arbor, MI, 1985. Bounded analytic functions. MR 779463
N. Danikas and Chr. Mouratides, Blaschke products in $Q_p$ spaces, Complex Variables Theory Appl. 43 (2000), no. 2, 199–209. MR 1812465, DOI 10.1080/17476930008815311
Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
Daniel Girela and José Ángel Peláez, On the membership in Bergman spaces of the derivative of a Blaschke product with zeros in a Stolz domain, Canad. Math. Bull. 49 (2006), no. 3, 381–388. MR 2252260, DOI 10.4153/CMB-2006-038-x
Daniel Girela, José Ángel Peláez, and Dragan Vukotić, Integrability of the derivative of a Blaschke product, Proc. Edinb. Math. Soc. (2) 50 (2007), no. 3, 673–687. MR 2360523, DOI 10.1017/S0013091504001014
Daniel Girela, José Ángel Peláez, and Dragan Vukotić, Interpolating Blaschke products: Stolz and tangential approach regions, Constr. Approx. 27 (2008), no. 2, 203–216. MR 2336422, DOI 10.1007/s00365-006-0651-6
Alan Gluchoff, The mean modulus of a Blaschke product with zeroes in a nontangential region, Complex Variables Theory Appl. 1 (1983), no. 4, 311–326. MR 706988, DOI 10.1080/17476938308814022
Janne Gröhn and Artur Nicolau, Inner functions in certain Hardy-Sobolev spaces, J. Funct. Anal. 272 (2017), no. 6, 2463–2486. MR 3603305, DOI 10.1016/j.jfa.2016.12.001
Haakan Hedenmalm, Boris Korenblum, and Kehe Zhu, Theory of Bergman spaces, Graduate Texts in Mathematics, vol. 199, Springer-Verlag, New York, 2000. MR 1758653, DOI 10.1007/978-1-4612-0497-8
Miroljub Jevtić, Blaschke products in Lipschitz spaces, Proc. Edinb. Math. Soc. (2) 52 (2009), no. 3, 689–705. MR 2546639, DOI 10.1017/S001309150700065X
Javad Mashreghi, Derivatives of inner functions, Fields Institute Monographs, vol. 31, Springer, New York; Fields Institute for Research in Mathematical Sciences, Toronto, ON, 2013. MR 2986324, DOI 10.1007/978-1-4614-5611-7
Miodrag Mateljević and Miroslav Pavlović, On the integral means of derivatives of the atomic function, Proc. Amer. Math. Soc. 86 (1982), no. 3, 455–458. MR 671214, DOI 10.1090/S0002-9939-1982-0671214-X
Miroslav Pavlović, Introduction to function spaces on the disk, Posebna Izdanja [Special Editions], vol. 20, Matematički Institut SANU, Belgrade, 2004. MR 2109650
José Ángel Peláez and Jouni Rättyä, Embedding theorems for Bergman spaces via harmonic analysis, Math. Ann. 362 (2015), no. 1-2, 205–239. MR 3343875, DOI 10.1007/s00208-014-1108-5
Fernando Pérez-González and Jouni Rättyä, Derivatives of inner functions in weighted Bergman spaces and the Schwarz-Pick lemma, Proc. Amer. Math. Soc. 145 (2017), no. 5, 2155–2166. MR 3611328, DOI 10.1090/proc/13384
F. Pérez-González, J. Rättyä, and A. Reijonen, Derivatives of inner functions in Bergman spaces induced by doubling weights, Ann. Acad. Sci. Fenn. Math. 42 (2017), 735-753.