An operator-valued multi-variable Poisson type integral is studied. In Section 2 we obtain a new equivalent condition for the existence of a so-called regular unitary dilation of an n-tuple T=(T₁,...,Tₙ) of commuting contractions. Our development in Section 2 also contains a new proof of the classical dilation result of S. Brehmer, B. Sz.-Nagy and I. Halperin. In Section 3 we turn to the boundary behavior of this operator-valued Poisson integral. The results obtained in this section improve upon...

We consider the following problem: find on ${ℂ}^2$ a plurisubharmonic function with a given order function. In particular, we prove that any positive ambiguous function on $ℂ{\mathbb{P}}^1$ which is constant outside a polar set is the order function of a plurisubharmonic function.

We study the sequence $u_n$, which is solution of $-div\left(a(x,𝔻u_n)\right)+{\Phi }^{\text{'}\text{'}}\left(\right|u_n\left|\right)u_n=f_n+g_n$ in $\Omega$ an open bounded set of ${𝐑}^N$ and $u_n=0$ on $\partial \Omega$, when $f_n$ tends to a measure concentrated on a set of null Orlicz-capacity. We consider the relation between this capacity and the $N$-function $\Phi$, and prove a non-existence result.

We study the sequence un, which is solution of $-\mathrm{div}\left(a(x,\nabla u_n)\right)+{\Phi }^{\text{'}\text{'}}\left(\right|u_n\left|\right)u_n=f_n+g_n$ in Ω an open bounded set of RN and un= 0 on ∂Ω, when fn tends to a measure concentrated on a set of null Orlicz-capacity. We consider the relation between this capacity and the N-function Φ, and prove a non-existence result.