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

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]**Axler, S., Bourdon, P. and Ramey, W.,*Harmonic Function Theory*. Springer-Verlag, New York-Berlin, 1992. MR**93f:31001****[AS]**Ahlfors, L. V. and Sario, L.,*Riemann Surfaces*. Princeton University Press, Princeton, New Jersey, 1960. MR**22:5729****[B]**Bourdon, P. S., Rudin's orthogonality problem and the Nevanlinna counting function.*Proc. Amer. Math. Soc***125**, 4(1997), 1187-1192. MR**98b:30034****[C1]**Carleson, L., An interpolation theorem for bounded analytic functions.*Amer. J. Math***80**(1958) 921-930. MR**32:8129****[C2]**Carleson, L., Interpolations by bounded analytic functions and the corona problem.*Ann. of Math.*,**76**(1962), 547-559. MR**25:5186****[C]**Conway, J. B.,*Functions of one complex variable*. Springer-Verlag, New York-Berlin, 1978. MR**80c:30003****[D]**Duren, P. L.,*Theory of**spaces*. Academic Press, New York, 1970. MR**42:3552****[F]**Fisher, S. D.,*Function Theory on Planar Domains*. John Wiley and Sons, New York, 1983. MR**85d:30001****[G]**Garnett, J. B.,*Bounded Analytic Functions*. Academic Press, New York, 1981. MR**83g:30037****[K]**Kakutani, S., Two-dimensional Brownian motion and harmonic functions.*Proc. Imp. Acad. Tokyo*,**20**(1944), 706-714. MR**7:315b****[Ke]**Kellogg, O. D.,*Foundations of Potential Theory*. Dover, New York, 1953. MR**36:5369****[L]**Littlewood, J. E., On inequalities in the theory of functions.*Proc. London Math. Soc.*(2)**23**(1925), 481-519.**[P]**Petersen, K. E.,*Brownian Motion, Hardy Spaces, and Bounded Mean Oscillation*. Cambridge University Press, Cambridge, 1977. MR**58:31383****[PS]**Port, S. C. and Stone, C. J.,*Brownian Motion and Classical Potential Theory*. Academic Press, New York, 1978. MR**58:11459****[R]**Rudin, W., A generalization of a theorem of Frostman,*Math. Scand.*,**21**(1967), 136-143. MR**38:3463****[S]**Shapiro, J. H., The essential norm of a composition operator.*Ann. of Math.*,**125**(1987), 375-404. MR**88c:47058****[Sp]**Springer, G.,*Introduction to Riemann Surfaces*. Addison-Wesley, Reading, Massachusetts, 1957. MR**19:1169g**

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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