Measures induced by analytic functions and a problem of Walter Rudin

Author:
Carl Sundberg

Journal:
J. Amer. Math. Soc. **16** (2003), 69-90

MSC (2000):
Primary 30D50

Published electronically:
September 10, 2002

MathSciNet review:
1937200

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The measure induced by a bounded analytic function on the unit disk may be defined by , where is normalized Lebesgue measure on . We discuss the problem of characterizing such measures, and produce some necessary conditions which we conjecture are sufficient. Our main results are a construction showing that our conjectured sufficient conditions are sufficient for a measure to be weakly approximable by induced measures, and a construction of a function , not a constant multiple of an inner function, whose induced measure is rotationally symmetric. This function is not inner, but satisfies if , thus answering a question posed by Walter Rudin.

**[ABR]**Sheldon Axler, Paul Bourdon, and Wade Ramey,*Harmonic function theory*, Graduate Texts in Mathematics, vol. 137, Springer-Verlag, New York, 1992. MR**1184139****[AS]**Lars V. Ahlfors and Leo Sario,*Riemann surfaces*, Princeton Mathematical Series, No. 26, Princeton University Press, Princeton, N.J., 1960. MR**0114911****[B]**Paul S. Bourdon,*Rudin’s orthogonality problem and the Nevanlinna counting function*, Proc. Amer. Math. Soc.**125**(1997), no. 4, 1187–1192. MR**1363413**, 10.1090/S0002-9939-97-03694-0**[C1]**Dietrich Göhde,*Zum Prinzip der kontraktiven Abbildung*, Math. Nachr.**30**(1965), 251–258 (German). MR**0190718****[C2]**Lennart Carleson,*Interpolations by bounded analytic functions and the corona problem*, Ann. of Math. (2)**76**(1962), 547–559. MR**0141789****[C]**John B. Conway,*Functions of one complex variable*, 2nd ed., Graduate Texts in Mathematics, vol. 11, Springer-Verlag, New York-Berlin, 1978. MR**503901****[D]**Peter L. Duren,*Theory of 𝐻^{𝑝} spaces*, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR**0268655****[F]**Stephen D. Fisher,*Function theory on planar domains*, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1983. A second course in complex analysis; A Wiley-Interscience Publication. MR**694693****[G]**John B. Garnett,*Bounded analytic functions*, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR**628971****[K]**Shizuo Kakutani,*Two-dimensional Brownian motion and harmonic functions*, Proc. Imp. Acad. Tokyo**20**(1944), 706–714. MR**0014647****[Ke]**Oliver Dimon Kellogg,*Foundations of potential theory*, Reprint from the first edition of 1929. Die Grundlehren der Mathematischen Wissenschaften, Band 31, Springer-Verlag, Berlin-New York, 1967. MR**0222317****[L]**Littlewood, J. E., On inequalities in the theory of functions.*Proc. London Math. Soc.*(2)**23**(1925), 481-519.**[P]**Karl Endel Petersen,*Brownian motion, Hardy spaces and bounded mean oscillation*, Cambridge University Press, Cambridge-New York-Melbourne, 1977. London Mathematical Society Lecture Note Series, No. 28. MR**0651556****[PS]**Sidney C. Port and Charles J. Stone,*Brownian motion and classical potential theory*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. Probability and Mathematical Statistics. MR**0492329****[R]**Walter Rudin,*A generalization of a theorem of Frostman*, Math. Scand**21**(1967), 136–143 (1968). MR**0235151****[S]**Joel H. Shapiro,*The essential norm of a composition operator*, Ann. of Math. (2)**125**(1987), no. 2, 375–404. MR**881273**, 10.2307/1971314**[Sp]**George Springer,*Introduction to Riemann surfaces*, Addison-Wesley Publishing Company, Inc., Reading, Mass., 1957. MR**0092855**

Retrieve articles in *Journal of the American Mathematical Society*
with MSC (2000):
30D50

Retrieve articles in all journals with MSC (2000): 30D50

Additional Information

**Carl Sundberg**

Affiliation:
Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300

Email:
sundberg@math.utk.edu

DOI:
https://doi.org/10.1090/S0894-0347-02-00404-6

Received by editor(s):
May 5, 2000

Received by editor(s) in revised form:
August 5, 2002

Published electronically:
September 10, 2002

Additional Notes:
Research supported in part by the National Science Foundation

Article copyright:
© Copyright 2002
American Mathematical Society