Fock-Sobolev spaces and their Carleson measures

We study a class of holomorphic spaces F-p,F-m consisting of entire functions on C-n such that partial derivative(alpha) f is in the Fock space F-p for all multi-indices a with vertical bar alpha vertical bar <= m. We prove a useful Fourier characterization, namely, f is an element of F-p,F-m if and only if z(alpha) f(z) is in F-p for all alpha with vertical bar alpha vertical bar = m. We obtain duality and interpolation results for these spaces, including the interesting fact that. for 0 < p <= 1, (F-p,F-m)* = F-infinity,F-m. We also characterize Carleson measures for F-p,F-m in terms of simple polynomial growth conditions. (C) 2012 Elsevier Inc. All rights reserved.