Fractional Cauchy transforms, multipliers and Cesàro operators

Doubtsov, Evgueni

doi:10.1090/S0002-9939-09-10122-3

Fractional Cauchy transforms, multipliers and Cesàro operators
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Proc. Amer. Math. Soc. 138 (2010), 663-673 Request permission

Abstract:

Let $B_n$ denote the unit ball in ${\mathbb C}^n$, $n\ge 1$. Given an $\alpha >0$, let ${\mathcal K}_\alpha (n)$ denote the class of functions defined for $z\in B_n$ by integrating the kernel $(1- \langle z, \zeta \rangle )^{-\alpha }$ against a complex Borel measure on the sphere $\{\zeta \in {\mathbb C}^n:\ |\zeta |=1\}$. We study properties of the holomorphic functions $g$ such that $fg\in {\mathcal K}_\alpha (n)$ for all $f\in {\mathcal K}_\alpha (n)$. Also, we investigate extended Cesàro operators on ${\mathcal K}_\alpha (n)$.

References

Alexandru Aleman and Aristomenis G. Siskakis, An integral operator on $H^p$, Complex Variables Theory Appl. 28 (1995), no. 2, 149–158. MR 1700079, DOI 10.1080/17476939508814844
Alexandru Aleman and Aristomenis G. Siskakis, Integration operators on Bergman spaces, Indiana Univ. Math. J. 46 (1997), no. 2, 337–356. MR 1481594, DOI 10.1512/iumj.1997.46.1373
Frank Beatrous and Jacob Burbea, Holomorphic Sobolev spaces on the ball, Dissertationes Math. (Rozprawy Mat.) 276 (1989), 60. MR 1010151
Joseph A. Cima, Alec L. Matheson, and William T. Ross, The Cauchy transform, Mathematical Surveys and Monographs, vol. 125, American Mathematical Society, Providence, RI, 2006. MR 2215991, DOI 10.1090/surv/125
Joseph A. Cima and Aristomenis G. Siskakis, Cauchy transforms and Cesàro averaging operators, Acta Sci. Math. (Szeged) 65 (1999), no. 3-4, 505–513. MR 1737267
E. Doubtsov, Growth spaces on circular domains: composition operators and Carleson measures, C. R. Math. Acad. Sci. Paris 347 (2009), no. 11-12, 609–612.
Evgueni Doubtsov, Multipliers of fractional Cauchy transforms, Integral Equations Operator Theory 64 (2009), no. 2, 177–192. MR 2516277, DOI 10.1007/s00020-009-1681-2
E. S. Dubtsov, Families of fractional Cauchy transforms in the ball, Algebra i Analiz 21 (2009) (Russian), to appear.
R. A. Hibschweiler and T. H. MacGregor, Multipliers of families of Cauchy-Stieltjes transforms, Trans. Amer. Math. Soc. 331 (1992), no. 1, 377–394. MR 1120775, DOI 10.1090/S0002-9947-1992-1120775-6
Rita A. Hibschweiler and Thomas H. MacGregor, Fractional Cauchy transforms, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 136, Chapman & Hall/CRC, Boca Raton, FL, 2006. MR 2189294
Zhangjian Hu, Extended Cesàro operators on mixed norm spaces, Proc. Amer. Math. Soc. 131 (2003), no. 7, 2171–2179. MR 1963765, DOI 10.1090/S0002-9939-02-06777-1
Donghan Luo and Thomas MacGregor, Multipliers of fractional Cauchy transforms and smoothness conditions, Canad. J. Math. 50 (1998), no. 3, 595–604. MR 1629835, DOI 10.4153/CJM-1998-033-3
Ch. Pommerenke, Schlichte Funktionen und analytische Funktionen von beschränkter mittlerer Oszillation, Comment. Math. Helv. 52 (1977), no. 4, 591–602 (German). MR 454017, DOI 10.1007/BF02567392
Walter Rudin, Function theory in the unit ball of $\textbf {C}^{n}$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR 601594
S. A. Vinogradov, Properties of multipliers of integrals of Cauchy-Stieltjes type, and some problems of factorization of analytic functions, Mathematical programming and related questions (Proc. Seventh Winter School, Drogobych, 1974) Central Èkonom.-Mat. Inst. Akad. Nauk SSSR, Moscow, 1976, pp. 5–39 (Russian). MR 0586560
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