Hilbert-type operator induced by radial weight on Hardyspaces

Abstract

We consider the Hilbert-type operator defined by Hω(f)(z)=∫10f(t)(1z∫z0Bωt(u)du)ω(t)dt,

where {Bωζ}ζ∈D are the reproducing kernels of the Bergman space A2ω induced by a radial weight ω in the unit disc D. We prove that Hω is bounded on the Hardy space Hp, 1<p<∞

, if and only if sup0≤r<1ωˆ(r)ωˆ(1+r2)<∞,(†)

and sup0<r<1(∫r01ωˆ(t)pdt)1p(∫1r(ωˆ(t)1−t)p′dt)1p′<∞,

where ωˆ(r)=∫1rω(s)ds . We also prove that Hω:H1→H1 is bounded if and only if (†

) holds and supr∈[0,1)ωˆ(r)1−r(∫r0dsωˆ(s))<∞.

As for the case p=∞ , Hω is bounded from H∞ to \mathord \mathrm{BMOA}, or to the Bloch space, if and only if (†) holds. In addition, we prove that there does not exist radial weights ω such that Hω:Hp→Hp, 1≤p<∞, is compact and we consider the action of Hω on some spaces of analytic functions closely related to Hardy spaces.