Generalized Hilbert Operators on Bergman and Dirichlet Spaces of Analytic Functions

Sunanda Naik ; Karabi Rajbangshi

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 63 (2015), p. 227-235 / Harvested from The Polish Digital Mathematics Library

Résumé

Let f be an analytic function on the unit disk . We define a generalized Hilbert-type operator $_{a,b}$ by $_{a,b}(f)(z) = Γ(a+1)/Γ(b+1) ∫_{0}^{1} (f(t)(1-t)^{b})/((1-tz)^{a+1}) dt$, where a and b are non-negative real numbers. In particular, for a = b = β, $_{a,b}$ becomes the generalized Hilbert operator $_β$, and β = 0 gives the classical Hilbert operator . In this article, we find conditions on a and b such that $_{a,b}$ is bounded on Dirichlet-type spaces $S^{p}$, 0 < p < 2, and on Bergman spaces $A^{p}$, 2 < p < ∞. Also we find an upper bound for the norm of the operator $_{a,b}$. These generalize some results of E. Diamantopolous (2004) and S. Li (2009).

Détail

Publié le : 2015-01-01

Zbl 1337.30063

EUDML-ID : urn:eudml:doc:281137