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We consider the weighted Bergman spaces $${\mathcal {H}L^{2}(\mathbb {B}^{d}, \mu_{\lambda})}$$ , where we set $${d\mu_{\lambda}(z) = c_{\lambda}(1-|z|^2)^{\lambda} d\tau(z)}$$ , with τ being the hyperbolic volume measure. These spaces are nonzero if and only if λ > d. For 0 < λ ≤ d, spaces with the same formula for the reproducing kernel can be defined using a Sobolev-type norm. We...