Weighted composition operators from \(F(p,q,s)\) to Bloch type spaces on the unit ball. (English) Zbl 1163.47021

Let \(B\) denote the complex \(n\)-dimensional unit ball. For \(p>0\), \(s>0\), \(q>-(n+1)\), with \(q+s>-1\), let \(F(p,q,s)\) denote the space of holomorphic functions \(f\) in \(B\) such that \[ \sup_{a\in B}\int_B| \nabla f(z)| ^p(1-| z| ^2)^qg^s(z,a)\,dv(z)<\infty, \] where \(dv\) is volume measure and \(g\) is Green’s function on \(B\). For \(\alpha>0\), let \(B^\alpha\) denote the space of holomorphic functions in \(B\) such that \[ \sup_{a\in B}(1-| a| ^2)^\alpha| \nabla f(a)| <\infty. \] The paper characterizes boundedness and compactness for weighted composition operators \(W_{\phi,\psi}\) between \(F(p,q,s)\) and \(B^\alpha\). Here, \(\phi\) is a holomorphic self-map of \(B\), \(\psi\) is a holomorphic function on \(B\), and \(W_{\phi,\psi}f=\psi f\circ\phi\).
Reviewer: Kehe Zhu (Albany)


47B33 Linear composition operators
32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
46E15 Banach spaces of continuous, differentiable or analytic functions
Full Text: DOI arXiv


[1] Cima J. A., Proc. Amer. Math. Soc. 99 pp 477–
[2] DOI: 10.1017/S144678870000183X · doi:10.1017/S144678870000183X
[3] Cowen C. C., Contemporary Mathematics 213, in: Studies in Composition Operators (1998) · doi:10.1090/conm/213/02846
[4] Cowen C. C., Composition Operators on Spaces of Analytic Functions (1995) · Zbl 0873.47017
[5] D’Angelo J. P., Several Complex Variables and Geometry of Real Hypersurfaces (1993)
[6] Jiang L. J., Acta Math. Sci. Ser. B Engl. Ed. 23 pp 252–
[7] DOI: 10.1090/S0002-9939-1993-1152987-6 · doi:10.1090/S0002-9939-1993-1152987-6
[8] DOI: 10.1090/S0002-9947-1995-1273508-X · doi:10.1090/S0002-9947-1995-1273508-X
[9] DOI: 10.1112/S0024610700008875 · Zbl 0959.47016 · doi:10.1112/S0024610700008875
[10] DOI: 10.2140/pjm.1999.188.339 · Zbl 0932.30034 · doi:10.2140/pjm.1999.188.339
[11] DOI: 10.1007/978-1-4613-8098-6 · doi:10.1007/978-1-4613-8098-6
[12] DOI: 10.1090/S0002-9939-1987-0883400-9 · doi:10.1090/S0002-9939-1987-0883400-9
[13] DOI: 10.1007/978-1-4612-0887-7 · doi:10.1007/978-1-4612-0887-7
[14] DOI: 10.1007/s101149900028 · Zbl 0967.32007 · doi:10.1007/s101149900028
[15] DOI: 10.1112/blms/12.4.241 · Zbl 0416.32010 · doi:10.1112/blms/12.4.241
[16] DOI: 10.1216/rmjm/1021477265 · Zbl 0978.32002 · doi:10.1216/rmjm/1021477265
[17] Zhao R. H., Ann. Acad. Sci. Fenn. Math. Diss. 105 pp 56–
[18] DOI: 10.1360/03ys9004 · Zbl 1217.32002 · doi:10.1360/03ys9004
[19] Zhou Z. H., J. Inequal. Appl. 2006 pp 22–
[20] DOI: 10.1007/BF02878708 · Zbl 1024.47010 · doi:10.1007/BF02878708
[21] DOI: 10.1307/mmj/1028575740 · Zbl 1044.47021 · doi:10.1307/mmj/1028575740
[22] Zhu K., Pure and Applied Mathematics 136, in: Operator Theory in Function Spaces (1990) · Zbl 0706.47019
[23] Zhu K., Graduate Texts in Mathematics 226, in: Spaces of Holomorphic functions in the Unit Ball (2004)
[24] Zhuo W. X., Acta Math. Sci. Ser. English Ed. 22 pp 295–
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