Strong pseudoconvexity in Banach spaces

Abstract

Having been unclear how to define that a domain is strictly pseudoconvex in the infinite-dimensional setting, we develop a general theory having Banach spaces in mind. We first focus on finite dimension and eliminate the need of two degrees of differentiability of the boundary of a domain, since differentiable functions are difficult to find in infinite dimension. We introduce $\ell$-strict pseudoconvexity for $\ell\geq 1$, $1$-strict pseudoconvexity at the boundary, $\ell$-uniform pseudoconvexity for $\ell\geq 0$ and finally strong pseudoconvexity. Defining $\ell$-strict pseudoconvexity and $\ell$-uniform pseudoconvexity for $\ell<2$ depends on extending a notion of strict plurisubharmonicity to cases lacking $C^2$-smoothness, first studying it in the sense of distribution and then considering it in infinite dimension. Examples of strictly plurisubharmonic functions as well as strongly pseudoconvex domains are presented, which end up related to important classical Banach spaces. Finally, some solutions to the inhomogeneous Cauchy-Riemann equations for $\overline{\partial}$-closed $(0,1)$-forms in infinite-dimensional domains are shown, giving new information about domains affinely isomorphic to the ball of $\ell_1$ which appeared in the study of strong pseudoconvexity and some more domains biholomorphically related to open and convex domains of $\ell_1$ such as its ball.