Hilbert-type operator induced by radial weight

2020/07/30 by Peláez, José Ángel, de la Rosa, Elena
#Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2007.15402

Abstract

We consider the Hilbert-type operator defined by H_ω(f)(z)=∫₀¹ f(t)((1)/(z)∫₀^z B^ω_t(u) du) ω(t)dt, where \B^ω_ζ\_ζ∈\mathbbD are the reproducing kernels of the Bergman space A²_ω induced by a radial weight ω in the unit disc \mathbbD. We prove that H_ω is bounded from H^∞ to the Bloch space if and only if ω belongs to the class \widehatD, which consists of radial weights ω satisfying the doubling condition sup_{0≤ r<1} \frac∫_r¹ ω(s) ds∫_(1+r)/(2)¹ω(s) ds

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