Molecular decompositions and interpolation (Q1128239)

Let \(D\) be a homogeneous Siegel domain of type II, \(B(\zeta,z)\) its Bergman kernel, and \(A^{p,r}(D)\) the subspace of holomorphic functions in \(L^p(D,B(z,z)^{-r} d\nu(z))\), where \(d\nu\) is the Lebesgue measure on \(D\). \textit{R. R. Coifman} and \textit{R. Rochberg} [Astérisque 77, 11-66 (1980; Zbl 0472.46040)] proved that if \(D\) is symmetric, then for each \(p\) there are constants \(C_1,C_2\) and points \(\zeta_i\in D\), \(i=1,2,\dots\), such that (i) each \(F\in A^{p,0}(D)\) can be represented as \[ F(z)=\sum_i \lambda_i [B(z,\zeta_i)^2/B(\zeta_i,\zeta_i)]^{1/p}, \quad\text{with}\quad \| \lambda\| _{l^p} \leq C_1 \| F\| _{A^{p,0}},\tag{*} \] and (ii) for any \(\{\lambda_i\}\in l^p\), the function \(F\) defined by (*) is in \(A^{p,0}(D)\) and \(\| F\| _{A^{p,0}}\leq C_2\| \lambda\| _{l^p}\). Further, \textit{R. Rochberg} [Mich. Math. J. 29, 229-236 (1982; Zbl 0496.32010)] showed that if \(S=\{\zeta_k\}\) is a sequence of points in \(D\) such that \(\inf_{i\neq j} d(\zeta_i,\zeta_j)=\eta>0\), where \(d\) denotes the Bergman distance, then the operator \(T\) defined by \(Tf(\zeta_k)=B(\zeta_k,\zeta_k)^{-1/p}f(\zeta_k)\) maps \(A^{p,0}(D)\) into \(l^p(S)\) and there exists \(\eta_0>0\) such that \(T\) possesses a right inverse when \(\eta>\eta_0\). In either case, analogous results were also established for \(A^{p,r}(D)\). In the present paper, these results are extended in two ways: first, more general weighted Bergman spaces \(A^{p,\rho}(D)\) are considered, with \(\rho\) a vector in \(\mathbf R^l\) where \(l\) is the rank of \(D\), of which \(A^{p,r}(D)\) are the special case \(\rho=(r,r,\dots,r)\); and second, apart from symmetric domains \(D\) they are also established for two non-symmetric homogeneous Siegel domains of type II, namely the tube domain \(D_0=\mathbf R^5+iV_0\) over the cone \(V_0\) of real, symmetric, positive definite \(3\times 3\) matrices \(M\) whose \((2,3)\)-entry is \(0\), and the Pyatecki-Shapiro domain \(D_1=\{(z,u)\in\mathbf C^3\times\mathbf C: \text{Im} z_3>0, (\text{Im }z_1-| u| ^2)\text{Im }z_3 -(\text{Im }z_2)^2>0\}\). The proofs use integral formulas and \(L^p\)-estimates for weighted Bergman projections obtained in the authors' earlier paper [Stud. Math. 115, No. 3, 219-239 (1995; Zbl 0842.32016)] and a generalization to \(D_0\) and \(D_1\) of a lemma due to Koranyi.