# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Carleson measures and a class of generalized integration operators on the Bergman space. (English) Zbl 1225.47032
Summary: We consider a linear operator $$I^{(n)}_{h,\varphi}f(z)=\int^z_0 f^{(n)}(\varphi(\xi))h(\zeta)\,d\zeta$$ induced by holomorphic maps $h$ and $\varphi$ of the open unit disk $\Bbb D$, where $\varphi({\Bbb D})\subset\Bbb D$ and $n$ is a non-negative integer. A complete characterization of when $I^{(n)}_{h,\varphi}$ is bounded on the Bergman space ${\cal A}^2$ is established by using Luecking’s result for Carleson measures. We also compute upper and lower bounds for the essential norm of this operator on the Bergman space.

##### MSC:
 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators 46E20 Hilbert spaces of continuous, differentiable or analytic functions 30H20 Bergman spaces, Fock spaces
Full Text:
##### References:
 [1] A. Aleman and J.A. Cima, An integral operator on $H^p$ and Hardy inequality , J. Anal. Math. 85 (2001), 157-176. · Zbl 1061.30025 · doi:10.1007/BF02788078 [2] A. Aleman and A.G. Siskakis, An integral operator on $H^p$ , Complex Variables 28 (1995), 149-158; 46 (1997), 337-356. · Zbl 0837.30024 [3] ---, Integration operators on Bergman spaces , Indiana Univ. Math. J. 46 (1997), 337-356. · Zbl 0951.47039 [4] C.C. Cowen and B.D. MacCluer, Composition operators on spaces of analytic functions , CRC Press, Boca Raton, 1995. · Zbl 0873.47017 [5] H. Hedenmalm, B. Korenblum and K. Zhu, Theory of Bergman spaces , Springer-Verlag, New York, 2000. · Zbl 0955.32003 [6] D.H. Luecking, Forward and reverse Carleson inequalities for functions in Bergman spaces and their derivatives , Amer. Math. J. 40 (1985), 85-111. · Zbl 0584.46042 · doi:10.2307/2374458 [7] Ch. Pommerenke, Schlichte Funktionen und analytische Funktionen von beschrankter mittlerer Oszillation , Comment. Math. Helv. 52 (1977), 591-602. · Zbl 0369.30012 · doi:10.1007/BF02567392 · eudml:139717 [8] J. Rättyä, Integration operator acting on Hardy and weighted Bergman spaces , Bull. Austral. Math. Soc. 75 (2007), 431-446. · Zbl 1197.30018 · doi:10.1017/S0004972700039356 [9] J.H. Shapiro, Composition operators and classical function theory , Springer-Verlag, New York, 1993. · Zbl 0791.30033 [10] A.G. Siskakis and R. Zhao, A Volterra type operator on spaces of analytic functions , Contemp. Math. 232 (1999), 299-311. · Zbl 0955.47029 [11] J. Xiao, Cesaro operators on Hardy, BMOA and Bloch spaces , Arch. Math. 68 (1997), 398-406. · Zbl 0870.30026 · doi:10.1007/s000130050072 [12] ---, Riemann-Stieltjes operators on weighted Bloch and Bergman spaces of the unit ball , J. London Math. Soc. 70 (2004), 199-214. · Zbl 1064.47034 · doi:10.1112/S0024610704005484 [13] R. Yoneda, Pointwise multipliers from $BMOA^{\alpha}$ to $BMOA^{\beta}$ , Complex Variables 49 (2004), 1045-1061. · Zbl 1084.47026 · doi:10.1080/02781070412331320448 [14] K. Zhu, Operator theory in function spaces , Marcel Dekker, New York, 1990.