We consider the family P of n-tuples P consisting of
polynomials P1β,β¦,Pnβ with nonnegative coefficients which satisfy
βiβPjβ(0)=Ξ΄i,jβ,i,j=1,β¦,n. With any such P, we
associate a Reinhardt domain β³Pβnβ that we will call the
generalized Hartogs triangle. We are particularly interested in the choices
Paβ=(P1,aβ,β¦,Pn,aβ),aβ₯0, where Pj,aβ(z)=zjβ+aβk=1nβzkβ,Β j=1,β¦,n. The generalized Hartogs triangle
associated with Paβ is given by \begin{equation} \triangle^{\!n}_a = \Big\{z
\in \mathbb C \times \mathbb C^{n-1}_* : |z_j|^2 < |z_{j+1}|^2(1-a|z_1|^2),
~j=1, \ldots, n-1,
|z_n|^2 + a|z_1|^2 < 1\Big\}.
\end{equation} The domain β³Pβnβ,nβ₯2 is never
polynomially convex. However, β³Pβnβ is always holomorphically
convex. With any PβP and mβNn, we associate a
positive semi-definite kernel KP,mββ on β³Pβnβ.
This combined with the Moore's theorem yields a reproducing kernel Hilbert
space Hm2β(β³Pβnβ) of holomorphic functions on
β³Pβnβ. We study the space Hm2β(β³Pβnβ) and the multiplication n-tuple Mzβ
acting on Hm2β(β³Pβnβ). It turns out that Mzβ is never rationally cyclic. Although the dimension of the joint kernel of
MzβββΞ» is constant of value 1 for every Ξ»ββ³Pβnβ, it has jump discontinuity at the serious singularity 0
of the boundary of β³Pβnβ with value equal to β. We
capitalize on the notion of joint subnormality to define a Hardy space on
β³0βnβ. This in turn gives an analog of the von Neumann's
inequality for β³0βnβ.Comment: Revised version with a figure; 42 page