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

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

Published electronically:
September 10, 2002

MathSciNet review:
1937200

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The measure $\mu _\varphi$ induced by a bounded analytic function $\varphi$ on the unit disk $U$ may be defined by $\mu _\varphi (E)=m(\varphi ^{-1}(E))$, where $m$ is normalized Lebesgue measure on $\partial U$. 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 $\varphi$, not a constant multiple of an inner function, whose induced measure is rotationally symmetric. This function is not inner, but satisfies $\int \varphi \left (e^{i\theta }\right )^m\overline {\varphi \left (e^{i\theta }\right )^n} \frac {d\theta }{2\pi }=0$ if $m\ne n$, thus answering a question posed by Walter Rudin.

- Sheldon Axler, Paul Bourdon, and Wade Ramey,
*Harmonic function theory*, Graduate Texts in Mathematics, vol. 137, Springer-Verlag, New York, 1992. MR**1184139** - Lars V. Ahlfors and Leo Sario,
*Riemann surfaces*, Princeton Mathematical Series, No. 26, Princeton University Press, Princeton, N.J., 1960. MR**0114911** - Paul S. Bourdon,
*Rudin’s orthogonality problem and the Nevanlinna counting function*, Proc. Amer. Math. Soc.**125**(1997), no. 4, 1187–1192. MR**1363413**, DOI https://doi.org/10.1090/S0002-9939-97-03694-0 - Dietrich Göhde,
*Zum Prinzip der kontraktiven Abbildung*, Math. Nachr.**30**(1965), 251–258 (German). MR**190718**, DOI https://doi.org/10.1002/mana.19650300312 - Lennart Carleson,
*Interpolations by bounded analytic functions and the corona problem*, Ann. of Math. (2)**76**(1962), 547–559. MR**141789**, DOI https://doi.org/10.2307/1970375 - John B. Conway,
*Functions of one complex variable*, 2nd ed., Graduate Texts in Mathematics, vol. 11, Springer-Verlag, New York-Berlin, 1978. MR**503901** - Peter L. Duren,
*Theory of $H^{p}$ spaces*, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR**0268655** - 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** - 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]bK Kakutani, S., Two-dimensional Brownian motion and harmonic functions. - Oliver Dimon Kellogg,
*Foundations of potential theory*, Die Grundlehren der Mathematischen Wissenschaften, Band 31, Springer-Verlag, Berlin-New York, 1967. Reprint from the first edition of 1929. MR**0222317**
[L]bL Littlewood, J. E., On inequalities in the theory of functions. - 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** - 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** - Walter Rudin,
*A generalization of a theorem of Frostman*, Math. Scand.**21**(1967), 136–143 (1968). MR**235151**, DOI https://doi.org/10.7146/math.scand.a-10853 - Joel H. Shapiro,
*The essential norm of a composition operator*, Ann. of Math. (2)**125**(1987), no. 2, 375–404. MR**881273**, DOI https://doi.org/10.2307/1971314
[Sp]bSp Springer, G.,

*Proc. Imp. Acad. Tokyo*,

**20**(1944), 706–714.

*Proc. London Math. Soc.*(2)

**23**(1925), 481–519.

*Introduction to Riemann Surfaces*. Addison-Wesley, Reading, Massachusetts, 1957.

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

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