Abstract:

Let f be a holomorphic function on the unit polydisc Dⁿ, with Taylor expan-
sion
f(z) = ∑^∞_|k|=0 a_kz_k ≡ ∑^∞_{k1+···+kn=0} a_{k1,···, knz^k11}· · · z^kn_n
where k = (k₁, · · · , k_n) ∈ Zⁿ₊. The authors define generalized Hilbert operator on Dⁿ by
H_γ,n(f)(z) ₌∑^∞_{|k|=0 i1,···,in≥0} ai₁,···,i_n Πⁿ_j=1Γ( γ_j + k_j + 1)Γ(k_j + i_j + 1) /Γ(k_j + 1)Γ(k_j + i_j +γ_j + 2)z^k,
where γ ∈ Cⁿ, such that Rγ_j > -1, j = 1, 2, · · · , n. An upper bound for the norm of the operator on Hardy spaces H^p(Dⁿ) is found. The authors also present a Fejér-Riesz type inequality on the weighted Bergman space on Dⁿ and find an invariant space for the generalized Hilbert operator.

Key words: Generalized Hilbert operator, Fejér-Riesz inequality, α-Bloch space