Riemann–Stieltjes operators from \(F(p,q,s)\) spaces to \(\alpha\)-Bloch spaces on the unit ball. (English) Zbl 1131.47030

Li, Songxiao

J. Inequal. Appl. 2006, Article ID 27874, 14 p. (2006).

Summary: Let \(H(B)\) denote the space of all holomorphic functions on the unit ball \(B\subset\mathbb C^n\). We investigate the following integral operators: \[ \begin{aligned} T_g(f)(z)&= \int_0^1 f(tz)\operatorname{Re}\,g(tz) (dt/t),\\ L_g(f)(z)&= \int_0^1 \operatorname{Re}\,f(tz)g(tz)(dt/t), \end{aligned} \] \(f\in H(B)\), \(z\in B\), where \(g\in H(B)\), and \(\operatorname{Re}\,h(z)= \sum_{j=1}^n z_j(\partial h/\partial z_j)(z)\) is the radial derivative of \(h\). The operator \(T_g\) can be considered as an extension of the Cesàro operator on the unit disk. The boundedness of two classes of Riemann–Stieltjes operators from general function space \(F(p,q,s)\), which includes Hardy space, Bergman space, \(Q_p\) space, BMOA space, and Bloch space, to \(\alpha\)-Bloch space \({\mathcal B}^\alpha\) in the unit ball is discussed in this paper.

MSC:

47B38	Linear operators on function spaces (general)
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

Keywords:

boundedness; Riemann-Stieltjes operators; Hardy space; Bergman space; BMOA space; Bloch space; \(\alpha\)-Bloch space

References:

[1]	Aleman, A; Cima, JA, An integral operator on[inlineequation not available: see fulltext.] and Hardy’s inequality, Journal d’Analyse Mathématique, 85, 157-176, (2001) · Zbl 1061.30025
[2]	Aleman, A; Siskakis, AG, An integral operator on[inlineequation not available: see fulltext.], Complex Variables. Theory and Application, 28, 149-158, (1995) · Zbl 0837.30024
[3]	Aleman, A; Siskakis, AG, Integration operators on Bergman spaces, Indiana University Mathematics Journal, 46, 337-356, (1997) · Zbl 0951.47039
[4]	Hardy, GH; Littlewood, JE, Some properties of fractional integrals. II, Mathematische Zeitschrift, 34, 403-439, (1932) · Zbl 0003.15601
[5]	Hu, ZJ, Extended Cesàro operators on mixed norm spaces, Proceedings of the American Mathematical Society, 131, 2171-2179, (2003) · Zbl 1054.47023
[6]	Hu, ZJ, Extended Cesáro operators on the Bloch space in the unit ball of[inlineequation not available: see fulltext.], Acta Mathematica Scientia. Series B. English Edition, 23, 561-566, (2003) · Zbl 1044.47023
[7]	Hu, ZJ, Extended Cesàro operators on Bergman spaces, Journal of Mathematical Analysis and Applications, 296, 435-454, (2004) · Zbl 1072.47029
[8]	Miao, J, The Cesàro operator is bounded on[inlineequation not available: see fulltext.] for[inlineequation not available: see fulltext.], Proceedings of the American Mathematical Society, 116, 1077-1079, (1992) · Zbl 0787.47029
[9]	Ouyang, C; Yang, W; Zhao, R, Möbius invariant[inlineequation not available: see fulltext.] spaces associated with the Green’s function on the unit ball of[inlineequation not available: see fulltext.], Pacific Journal of Mathematics, 182, 69-99, (1998) · Zbl 0893.32005
[10]	Pommerenke, C, Schlichte funktionen und analytische funktionen von beschränkter mittlerer oszillation, Commentarii Mathematici Helvetici, 52, 591-602, (1977) · Zbl 0369.30012
[11]	Shi, J-H; Ren, G-B, Boundedness of the Cesàro operator on mixed norm spaces, Proceedings of the American Mathematical Society, 126, 3553-3560, (1998) · Zbl 0905.47019
[12]	Siskakis, AG, Composition semigroups and the Cesàro operator on[inlineequation not available: see fulltext.], Journal of the London Mathematical Society. Second Series, 36, 153-164, (1987) · Zbl 0634.47038
[13]	Siskakis, AG, The Cesàro operator is bounded on[inlineequation not available: see fulltext.], Proceedings of the American Mathematical Society, 110, 461-462, (1990) · Zbl 0719.47020
[14]	Siskakis, AG; Zhao, R, A Volterra type operator on spaces of analytic functions, No. 232, 299-311, (1999), Rhode Island · Zbl 0955.47029
[15]	Stević, S, On an integral operator on the unit ball in[inlineequation not available: see fulltext.], Journal of Inequalities and Applications, 2005, 81-88, (2005) · Zbl 1074.47013
[16]	Xiao, J, Cesàro-type operators on Hardy, BMOA and Bloch spaces, Archiv der Mathematik, 68, 398-406, (1997) · Zbl 0870.30026
[17]	Xiao, J, Riemann-Stieltjes operators on weighted Bloch and Bergman spaces of the unit ball, Journal of the London Mathematical Society. Second Series, 70, 199-214, (2004) · Zbl 1064.47034
[18]	Yoneda, R, Pointwise multipliers from[inlineequation not available: see fulltext.] to[inlineequation not available: see fulltext.], Complex Variables. Theory and Application, 49, 1045-1061, (2004) · Zbl 1084.47026
[19]	Zhang, XJ, Multipliers on some holomorphic function spaces, Chinese Annals of Mathematics. Series A, 26, 477-486, (2005) · Zbl 1084.32002
[20]	Zhao R: On a general family of function spaces.Annales Academiae Scientiarum Fennicae. Mathematica. Dissertationes 1996, (105):56.
[21]	Zhu K: Spaces of Holomorphic Functions in the Unit Ball, Graduate Texts in Mathematics. Volume 226. Springer, New York; 2005:x+271. · Zbl 1067.32005