Maximum boundary regularity of bounded Hua-harmonic functions on tube domains (Q1886945)

Let \(U^n\subset \mathbb C^{n+1}\) be the Siegel upper half-plane and \(F\) the Poisson-Szegö integral. It is known that if \(F\) has bounded transversal Euclidean derivatives up to the boundary, then \(F\) is pluriharmonic. The goal of this article is to extend this statement. To state the result we have to introduce some notation. Let \(D=V+\imath \Omega\) be an irreducible symmetric Siegel domain of tube type, \(V\) being an \(m\)-dimensional real Euclidean space and \(\Omega\) an irreducible symmetric cone inside \(V\). Let \(J\) be the complex structure and \(T^{1,0}\) the eigenspace of \(J\) such that \(J\bigl| _{T^{1,0}}=\imath Id\). Next, let \(\nabla\) be the Riemannian connection induced by the Bergman metric on \(D\). Given complex vector fields \(Z,W\), denote by \[ R(Z,W)=\nabla _Z \nabla _W - \nabla _W \nabla _Z - \nabla _{[Z,W]} \] the curvature tensor restricted to \(T^{1,0}(D)\), and \(\Delta (Z,W)f=(Z\bar{W}-\nabla _Z \bar{W})f\). Finally, the Hua system is defined as \[ Hf=\sum _{j,k}(\Delta (E_j,E_k)f)R(\bar{E_j},E_k), \] where \(E_1,\ldots ,E_m\) is an orthonormal frame of \(T^{1,0}\) for the canonical Hermitian product associated to the Bergman metric, and \(H\)-harmonic functions are those annihilated by \(H\). Main Theorem. Let \(D\) be an irreducible symmetric domain of tube type. There exists \(k\) (depending on the dimension and the rank) such that, if \(F\) is bounded \(H\)-harmonic and has bounded derivatives up to the order \(k\), then \(F\) is pluriharmonic.