Spectral properties of the ∂̄-canonical solution operator

In this paper we consider a Hilbert space H(0,1) of (0,1)-forms and a space L2(μ) of square integrable functions with respect to a measure μ on a rotation invariant open set Ω in Cn. We give necessary and sufficient conditions, in terms of the moments of the measure μ, for the canonical solution operator of the ∂̄-equation to be bounded, compact and in the Schatten p-class from H(0,1) into L2(μ). Examples of H(0,1) can be chosen to be the space of (0,1)-forms with coefficients in one of the classical Hilbert spaces of holomorphic functions such as the weighted Bergman space, the Hardy space, the Hardy–Sobolev space or the Möbius invariant space.