Characterizations for Besov spaces and applications. I. (Q2568193)

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 \textit{F. Beatrous} and \textit{J. Burbea} [Diss. Math. 276 (1989; Zbl 0691.46024)] and of \textit{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.