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We study a class of holomorphic spaces F^p,m consisting of entire functions f on Cⁿ such that ∂^αf is in the Fock space F^p for all multi-indices α with |α|⩽m. We prove a useful Fourier characterization, namely, f∈F^p,m if and only if z^αf(z) is in F^p for all α with |α|=m. We obtain duality and interpolation results for these spaces, including the interesting fact that, for 0<p⩽1, [Formula: see text] . We also characterize Carleson measures for F^p,m in terms of simple polynomial growth conditions.