## Rocky Mountain Journal of Mathematics

### Regularity of extremal functions in weighted Bergman and Fock type spaces

Timothy Ferguson

#### Abstract

We discuss the regularity of extremal functions in certain weighted Bergman and Fock type spaces. Given an appropriate analytic function $k$, the corresponding extremal function is the function with unit norm maximizing $Re \int _\Omega f(z) \overline {k(z)}\, \nu (z) \, dA(z)$ over all functions $f$ of unit norm, where $\nu$ is the weight function and $\Omega$ is the domain of the functions in the space. We consider the case where $\nu (z)$ is a decreasing radial function satisfying some additional assumptions, and where $\Omega$ is either a disc centered at the origin or the entire complex plane. We show that, if $k$ grows slowly in a certain sense, then $f$ must grow slowly in a related sense. We also discuss a relation between the integrability and growth of certain log-convex functions and apply the result to obtain information about the growth of integral means of extremal functions in Fock type spaces.

#### Article information

Source
Rocky Mountain J. Math., Volume 49, Number 1 (2019), 47-71.

Dates
First available in Project Euclid: 10 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1552186951

Digital Object Identifier
doi:10.1216/RMJ-2019-49-1-47

Mathematical Reviews number (MathSciNet)
MR3921866

Zentralblatt MATH identifier
07036618

#### Citation

Ferguson, Timothy. Regularity of extremal functions in weighted Bergman and Fock type spaces. Rocky Mountain J. Math. 49 (2019), no. 1, 47--71. doi:10.1216/RMJ-2019-49-1-47. https://projecteuclid.org/euclid.rmjm/1552186951

#### References

• D. Aharonov, C. Bénéteau, D. Khavinson and H. Shapiro, Extremal problems for nonvanishing functions in Bergman spaces, Oper. Th. Adv. Appl. 158 (2005), 59–86.
• C. Bénéteau, B.J. Carswell and S. Kouchekian, Extremal problems in the Fock space, Comp. Meth. Funct. Th. 10 (2010), 189–206.
• C. Bénéteau and D. Khavinson, A survey of linear extremal problems in analytic function spaces, CRM Proc. Lect. Notes 55 (2012), 33–46.
• J.A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), 396–414.
• P. Duren, Theory of $H\sp{p}$ spaces, Pure Appl. Math. 38 (1970).
• P. Duren, D. Khavinson and H.S. Shapiro, Extremal functions in invariant subspaces of Bergman spaces, Illinois J. Math. 40 (1996), 202–210.
• P. Duren, D. Khavinson, H.S. Shapiro and C. Sundberg, Contractive zero-divisors in Bergman spaces, Pacific J. Math. 157 (1993), 37–56.
• ––––, Invariant subspaces in Bergman spaces and the biharmonic equation, Michigan Math. J. 41 (1994), 247–259.
• P. Duren and A. Schuster, Bergman spaces, Math. Surv. Mono. 100 (2004).
• T. Ferguson, Continuity of extremal elements in uniformly convex spaces, Proc. Amer. Math. Soc. 137 (2009), 2645–2653.
• ––––, Extremal problems in Bergman spaces and an extension of Ryabykh's theorem, Illinois J. Math. 55 (2011), 555–573.
• E. Fischer, Über algebraische Modulsysteme und lineare homogene partielle Differentialgleichungen mit konstanten Koeffizienten, J. reine angew. Math. 140 (1911), 48–82.
• ––––, Über die Differentiationsprozesse der Algebra, J. reine angew. Math. 148 (1918), 1–78.
• J. Hansbo, Reproducing kernels and contractive divisors in Bergman spaces, J. Math. Sci. 92 (1998), 3657–3674.
• H. Hedenmalm, A factorization theorem for square area-integrable analytic functions, J. reine angew. Math. 422 (1991), 45–68.
• H. Hedenmalm, B. Korenblum and K. Zhu, Theory of Bergman spaces, Grad. Texts Math. 199 (2000).
• D. Khavinson, J.E. McCarthy and H. Shapiro, Best approximation in the mean by analytic and harmonic functions, Indiana Univ. Math. J. 49 (2000), 1481–1513.
• D. Khavinson and M. Stessin, Certain linear extremal problems in Bergman spaces of analytic functions, Indiana Univ. Math. J. 46 (1997), 933–974.
• T.H. MacGregor and M.I. Stessin, Weighted reproducing kernels in Bergman spaces, Michigan Math. J. 41 (1994), 523–533.
• D.J. Newman and H.S. Shapiro, A Hilbert spaces of entire functions related to the operational calculus, University of Michigan, Ann Arbor (1964), 1–91 (mimeographic notes).
• ––––, Certain Hilbert spaces of entire functions, Bull. Amer. Math. Soc. 72 (1966), 971–977.
• ––––, Fischer spaces of entire functions, Proc. Sympos. Pure Math. (1968), 360–369.
• NIST digital library of mathematical functions, http://dlmf.nist.gov/, release 1.0.6 of 2013-05-06, online companion to 24.
• F.W.J. Olver, D.W. Lozier, R.F. Boisvert and C.W. Clark, eds., NIST handbook of mathematical functions, Cambridge University Press, New York, 2010, print companion to 23.
• V.G. Ryabykh, Extremal problems for summable analytic functions, Sibirsk. Mat. Zh. 27 (1986), 212–217, 226.
• H.S. Shapiro, Topics in approximation theory, Lect. Notes Math. 187 (1971).
• C. Sundberg, Analytic continuability of Bergman inner functions, Michigan Math. J. 44 (1997), 399–407.
• D. Vukotić, A sharp estimate for $A^p_\alpha$ functions in ${\bf C}^n$, Proc. Amer. Math. Soc. 117 (1993), 753–756.
• ––––, Linear extremal problems for Bergman spaces, Expo. Math. 14 (1996), 313–352.
• K. Zhu, Analysis on Fock spaces, Grad. Texts Math. 263 (2012).