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Register a new userEstimates for solutions of the $partial$-equation and application to the characterization of the zero varieties of the functions of the nevanlinna class for lineally convex domains of finite type
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In the late ten years, the resolution of the equation $barpartial u=f$ with sharp estimates has been intensively studied for convex domains of finite type by many authors. In this paper, we consider the case of lineally convex domains. As the method used to obtain global estimates for a support function cannot be carried out in this case, we use a kernel that does not gives directly a solution of the $barpartial$-equation but only a representation formula which allows us to end the resolution of the equation using Kohns $L^2$ theory. As an application we give the characterization of the zero sets of the functions of the Nevanlinna class for lineally convex domains of finite type.
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We obtain uniform estimates for the canonical solution to $barpartial u=f$ on the Cartesian product of smoothly bounded planar domains, when $f$ is continuous up to the boundary. This generalizes Landuccis result for the bidisc toward higher dimensional product domains.
Given a domain $Omega$ in $mathbb{C}^m$, and a finite set of points $z_1,ldots, z_nin Omega$ and $w_1,ldots, w_nin mathbb{D}$ (the open unit disc in the complex plane), the $Pick, interpolation, problem$ asks when there is a holomorphic function $f:Omega rightarrow overline{mathbb{D}}$ such that $f(z_i)=w_i,1leq ileq n$. Pick gave a condition on the data ${z_i, w_i:1leq ileq n}$ for such an $interpolant$ to exist if $Omega=mathbb{D}$. Nevanlinna characterized all possible functions $f$ that $interpolate$ the data. We generalize Nevanlinnas result to an arbitrary set $Omega$. In this case, the function $f$ comes from the Schur-Agler class. The abstract result is then applied to three examples - the bidisc, the symmetrized bidisc and the annulus. In these examples, the Schur-Agler class is the same as the Schur class.
We prove that the $mathcal{H}^p$-corona problem has a solution for convex domains of finite type in $mathbb{C}^n$, $n ge 2$.
In this paper we investigate finitely generated ideals in the Nevanlinna class. We prove analogues to some known results for the algebra of bounded analytic functions $H^{infty}$. We also show that, in contrast to the $H^{infty}$-case, the stable rank of the Nevanlinna class is strictly bigger than 1.
The Leray transform and related boundary operators are studied for a class of convex Reinhardt domains in $mathbb C^2$. Our class is self-dual; it contains some domains with less than $C^2$-smooth boundary and also some domains with smooth boundary and degenerate Levi form. $L^2$-regularity is proved, and essential spectra are computed with respect to a family of boundary measures which includes surface measure. A duality principle is established providing explicit unitary equivalence between operators on domains in our class and operators on the corresponding polar domains. Many of these results are new even for the classical case of smoothly bounded strongly convex Reinhardt domains.
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