Generalized Hilbert operator and Fejér-Riesz type inequalities on the polydisc

Abstract

Let f be a holomorphic function on the unit polydisc {A figure is presented}n, with Taylor expansion f (z) underover(∑, | k | = 0, ∞) ak zk ≡ underover(∑, k1 + ⋯ + kn = 0, ∞) ak1, ..., kn z1k1 ⋯ znkn where k = (k1, ⋯, kn) ∈ ℤ+n. The authors define generalized Hilbert operator on {A figure is presented}n by ℋγ, n (f) (z) = underover(∑, | k | = 0 i1, ⋯, in ≥ 0, ∞) ai1, ⋯, in underover(∏, j = 1, n) frac(Γ (γj + kj + 1) Γ (kj + ij + 1), Γ (kj + 1) Γ (kj + ij + γj + 2)) zk where γ ∈ ℂn, such that ℜγj > - 1, 2, ⋯, n. An upper bound for the norm of the operator on Hardy spaces ℍp({A figure is presented}n) is found. The authors also present a Fejér-Riesz type inequality on the weighted Bergman space on {A figure is presented}p and find an invariant space for the generalized Hilbert operator.