Regularity of the \(\bar {\partial }\)-Neumann problem at point of infinite type
The authors present a unified treatment of estimates for the \(\overline{\partial}\)-Neumann problem on \(q\)-pseudoconvex (resp. \(q\)-pseudoconcave) domains. This treatment covers in particular subelliptic, compactness, and superlogarithmic estimates. The main theme is most easily explained in the pseudoconvex case. The authors introduce a condition \((f-P-q)\), where \(f\) is a smooth monotone increasing function from \(\mathbb{R}^{+}\) to \(\mathbb{R}^{+}\) that satisfies \(f(t) \lesssim t^{1/2}\). The condition, modeled after Catlin's classical condition \((P)\), then says, somewhat roughly speaking, that there should exist bounded functions whose complex Hessian blows up at least like \(f(1/d(z))\), where \(d(z)\) is the boundary distance. One of the main results of the paper is now the following: \((f-P-q)\) implies the estimate \( \|f(\Lambda)u\|^{2} \lesssim \|\overline{\partial}u\|^{2} + \|\overline{\partial}^{*}u\|^{2} + \|u\|^{2}, \; u \in C^{\infty}(\overline{\Omega}) \cap dom(\overline{\partial}^{*})\), where \(\Lambda\) is the usual tangential Bessel potential of order1. (Actually, a local version is given.) Subelliptic, superlogarithmic, and compactness estimates correspond, respectively, to the choices for f (or a family of \(f\)'s in the latter two cases, since those estimates are really families of estimates) of \(f(t) \geq Ct^{\varepsilon}\), \(f_{k} \geq k\log(t),\;\forall k\), and \(f_{k} \geq k, \; \forall k\). In the case of \(q\)-pseudoconvex (\(q\)-pseudoconcave) domains, the definition of \((f-P-q)\) has to be suitably modified, and the above implication holds for a restricted set of form levels, as usual. In section 2, the authors discuss a geometric criterion that ensures that property \((f-P-q)\) holds. In section 3, they discuss several interesting classes of domains where their condition can be verified. Finally, in section 4 it is indicated how to adapt Kohn's microlocal methods for \(\overline{\partial}_{b}\) to accommodate the more general setup in the present paper.
scientific article; zbMATH DE number 5809707
scientific article