## Revista Matemática Iberoamericana

### Outer and inner vanishing measures and division in $H^\infty + C$

Keiji Izuchi

#### Abstract

Measures on the unit circle are well studied from the view of Fourier analysis. In this paper, we investigate measures from the view of Poisson integrals and of divisibility of singular inner functions in $H^\infty + C$. Especially, we study singular measures which have outer and inner vanishing measures. It is given two decompositions of a singular positive measure. As applications, it is studied division theorems in $H^\infty + C$.

#### Article information

Source
Rev. Mat. Iberoamericana, Volume 18, Number 3 (2002), 511-540.

Dates
First available in Project Euclid: 28 April 2003

https://projecteuclid.org/euclid.rmi/1051544318

Mathematical Reviews number (MathSciNet)
MR1954863

Zentralblatt MATH identifier
1034.46046

#### Citation

Izuchi, Keiji. Outer and inner vanishing measures and division in $H^\infty + C$. Rev. Mat. Iberoamericana 18 (2002), no. 3, 511--540. https://projecteuclid.org/euclid.rmi/1051544318

#### References

• Axler, S., Gorkin, P.: Divisibility in Douglas algebras. Michigan Math. J. 31 (1984), 89-94.
• Carleson, L.: Interpolation by bounded analytic functions and the corona problem. Ann. of Math. 76 (1962), 547-559.
• Chang, S.-Y.: A characterization of Douglas subalgebras. Acta Math. 137 (1976), 81-89.
• Garnett, J.: Bounded Analytic Functions. Academic Press, 1981.
• Gorkin, P.: Singular functions and division in $H^\infty + C$. Proc. Amer. Math. Soc. 92 (1984), 268-270.
• Guillory, C., Izuchi, K., Sarason, D.: Interpolating Blaschke products and division in Douglas algebras. Proc. Roy. Irish Acad. Sect. A 84 (1984), 1-7.
• Guillory, C., Sarason, D.: Division in $H^\infty + C$. Michigan Math. J. 28 (1981), 173-181.
• Hoffman, K.: Banach Spaces of Analytic Functions. Prentice Hall, Englewood Cliffs, 1962.
• Hoffman, K.: Bounded analytic functions and Gleason parts. Ann. of Math. (2) 86 (1967), 74-111.
• Izuchi, K.: Countably generated Douglas algebras. Trans. Amer. Math. Soc. 299 (1987), 171-192.
• Izuchi, K.: Factorization of Blaschke products. Michigan Math. J. 40 (1993), 53-75.
• Izuchi, K.: Interpolating Blaschke products and factorization theorems. J. London Math. Soc. (2) 50 (1994), 547-567.
• Izuchi, K.: Weak infinite powers of Blaschke products. J. Anal. Math. 75 (1998), 135-154.
• Izuchi, K.: Singular inner functions of $L^1$-type II. J. Math. Soc. Japan 53 (2001), 285-305.
• Izuchi, K.: Common zero sets of equivalent singular measures, preprint.
• Izuchi, K., Izuchi, Y.: Inner functions and division in Douglas algebras. Michigan Math. J. 33 (1986), 435-443.
• Izuchi, K., Niwa, N.: Singular inner functions of $L^1$-type. J. Korean Math. Soc. 36 (1999), 787-811.
• Marshall, D.: Approximation and interpolation by inner functions. Thesis, University of California, Los Angeles, 1976.
• Marshall, D.: Subalgebras od $L^\infty$ containing $H^\infty$. Acta Math. 137 (1976), 91-98.
• Sarason, D.: Algebras of functions on the unit circle. Bull. Amer. Math. Soc. 79 (1973), 286-299.
• Younis, R.: Division in Douglas algebras and some applications. Arch. Math. 45 (1985), 550-560.
• Ziskind, S.: Interpolating sequences and Shilov boundary of $H^\infty(\Delta)$. J. Funct. Anal. 21 (1976), 380-388.