This article is a contribution to the theory of holomorphic function spaces on domains \(D\) in \(\mathbb{C}^n\) for which the complex geometry of the boundary is well understood: namely, strongly pseudoconvex domains, domains of finite type in \(\mathbb{C}^2\), and convex domains of finite type in \(\mathbb{C}^n\).
Let \(P\) denote the Bergman projection from the space \(L^2(D)\) of square-integrable functions onto the holomorphic subspace, and let \(K(w,z)\) denote the corresponding Bergman kernel function. Let \(d\lambda(z)\) denote the Lebesgue measure weighted by the factor \(K(z,z)\), and let \(\delta(z)\) denote the distance from a point \(z\) in \(D\) to the boundary of \(D\). The authors consider the holomorphic Besov space \(B^p(D)\) consisting of those holomorphic functions \(f\) on \(D\) for which \(\delta(z)^{n+1}\nabla^{n+1} f(z)\) belongs to the space \(L^p(D,d\lambda)\); the space \(B^\infty(D)\) is understood as the usual holomorphic Bloch space.
The main theorem states that when \(1\leq p\leq\infty\), the Bergman projection \(P\) maps the space \(L^p(D,d\lambda)\) continuously onto \(B^p(D)\). Moreover, there is a concrete map \(V\colon B^p(D) \to L^p(D,d\lambda)\) which is bounded and linear such that \(PV\) equals the identity on \(B^p(D)\). For the case of the unit ball, the theorem is contained in work of F. Beatrous and J. Burbea [Diss. Math. 276 (1989; Zbl 0691.46024)] and of M. M. Peloso [Mich. Math. J. 39, No. 3, 509–536 (1992; Zbl 0779.32012)]. The authors give as applications some results on small Hankel operators.