Multipliers of families of Cauchy-Stieltjes transforms

Hibschweiler, R. A.; MacGregor, T. H.

doi:10.1090/S0002-9947-1992-1120775-6

Multipliers of families of Cauchy-Stieltjes transforms
HTML articles powered by AMS MathViewer

by R. A. Hibschweiler and T. H. MacGregor PDF

Trans. Amer. Math. Soc. 331 (1992), 377-394 Request permission

Abstract:

For $\alpha > 0$ let ${\mathcal {F}_\alpha }$ denote the class of functions defined for $|z| < 1$ by integrating $1/{(1 - xz)^\alpha }$ against a complex measure on $|x|= 1$. A function $g$ holomorphic in $|z| < 1$ is a multiplier of ${\mathcal {F}_\alpha }$ if $f \in {\mathcal {F}_\alpha }$ implies $gf \in {\mathcal {F}_\alpha }$. The class of all such multipliers is denoted by ${\mathcal {M}_\alpha }$. Various properties of ${\mathcal {M}_\alpha }$ are studied in this paper. For example, it is proven that $\alpha < \beta$ implies ${\mathcal {M}_\alpha } \subset {\mathcal {M}_\beta }$, and also that ${\mathcal {M}_\alpha } \subset {H^\infty }$. Examples are given of bounded functions which are not multipliers. A new proof is given of a theorem of Vinogradov which asserts that if $f’$ is in the Hardy class ${H^1}$, then $f \in {\mathcal {M}_1}$. Also the theorem is improved to $f’ \in {H^1}$ implies $f \in {\mathcal {M}_\alpha }$, for all $\alpha > 0$. Finally, let $\alpha > 0$ and let $f$ be holomorphic in $|z| < 1$. It is known that $f$ is bounded if and only if its Cesàro sums are uniformly bounded in $|z| \leq 1$. This result is generalized using suitable polynomials defined for $\alpha > 0$.

References

Paul Bourdon and Joseph A. Cima, On integrals of Cauchy-Stieltjes type, Houston J. Math. 14 (1988), no. 4, 465–474. MR 998448
L. Brickman, D. J. Hallenbeck, T. H. MacGregor, and D. R. Wilken, Convex hulls and extreme points of families of starlike and convex mappings, Trans. Amer. Math. Soc. 185 (1973), 413–428 (1974). MR 338337, DOI 10.1090/S0002-9947-1973-0338337-5

The Taylor series

Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
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
D. J. Hallenbeck and T. H. MacGregor, Radial growth and variation of bounded analytic functions, Proc. Edinburgh Math. Soc. (2) 31 (1988), no. 3, 489–498. MR 969079, DOI 10.1017/S001309150003769X
V. P. Havin, On analytic functions representable by an integral of Cauchy-Stieltjes type, Vestnik Leningrad. Univ. Ser. Mat. Meh. Astr. 13 (1958), no. 1, 66–79 (Russian, with English summary). MR 0095256
V. P. Havin, Relations between certain classes of functions regular in the unit circle, Vestnik Leningrad. Univ. 17 (1962), no. 1, 102–110 (Russian, with English summary). MR 0152660
R. A. Hibschweiler and T. H. MacGregor, Closure properties of families of Cauchy-Stieltjes transforms, Proc. Amer. Math. Soc. 105 (1989), no. 3, 615–621. MR 938912, DOI 10.1090/S0002-9939-1989-0938912-8
S. V. Hruščev and S. A. Vinogradov, Inner functions and multipliers of Cauchy type integrals, Ark. Mat. 19 (1981), no. 1, 23–42. MR 625535, DOI 10.1007/BF02384467
Paul Koosis, Introduction to $H_{p}$ spaces, London Mathematical Society Lecture Note Series, vol. 40, Cambridge University Press, Cambridge-New York, 1980. With an appendix on Wolff’s proof of the corona theorem. MR 565451
T. H. MacGregor, Analytic and univalent functions with integral representations involving complex measures, Indiana Univ. Math. J. 36 (1987), no. 1, 109–130. MR 876994, DOI 10.1512/iumj.1987.36.36006
Walter Rudin, The radial variation of analytic functions, Duke Math. J. 22 (1955), 235–242. MR 79093
S. A. Vinogradov, M. G. Goluzina, and V. P. Havin, Multipliers and divisors of Cauchy-Stieltjes type integrals, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 19 (1970), 55–78 (Russian). MR 0291471

Properties of multipliers of Cauchy-Stieltjes integrals and some factorization problems for analytic functions

115

Lawrence Zalcman, Real proofs of complex theorems (and vice versa), Amer. Math. Monthly 81 (1974), 115–137. MR 328028, DOI 10.2307/2976953

Trigonometric series