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Generalized Hilbert Operators on Bergman and Dirichlet Spaces of Analytic Functions

Naik S., Rajbangshi K.

Bulletin of the Polish Academy of Sciences. Mathematics

2015

Vol. 63, no. 3

227--235

Let f be an analytic function on the unit disk D. We define a generalized Hilbert-type operator Ha, b by Ha, b (f)(z) = [WZÓR], where a and b are non-negative real numbers. In particular, for a=b=β, Ha, b becomes the generalized Hilbert operator Hβ, and β=0 gives the classical Hilbert operator H. In this article, we find conditions on a and b such that Ha, b is bounded on Dirichlet-type spaces Sp, 0 < p < 2, and on Bergman spaces Ap, 2 < p < ∞. Also we find an upper bound for the norm of the operator Ha, b. These generalize some results of E. Diamantopolous (2004) and S. Li (2009).

Weighted composition operators via Berezin transform and Carleson measure

Kumar R., Singh K. J.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2006

Vol. 46, [Z] 1

63-78

In this paper, we study the boundedness and the compactness of weighted composition operators on Hardy spaces and weighted Bergman spaces of the unit polydisc in C^n.

Exceptional sets for functions from the weighted Bergman spaces in the ball

Jakóbczak P.

Demonstratio Mathematica

2003

Vol. 36, nr 1

83--92

In this note we study the behaviour of holomorphic functions in the unit ball BN in CN on one-dimensional complex subspaces of CN . The behaviour of functions is described in terms of L2-integrability with weights on the sets L D BN, where L runs over different families E of one-dimensional complex subspaces of CN .