Strong pseudoconvexity in Banach spaces

Since it is unclear how to define a domain that is strictly pseudoconvex in the infinite-dimensional setting, we develop a general theory with Banach spaces in mind. We first focus on the finite dimensional case and eliminate the need for two degrees of differentiability of the boundary of a domain, since differentiable functions are difficult to find in infinite dimensions. We introduce \(\ell \) $\ell$ -strict pseudoconvexity for \(\ell \ge 1\) $\ell \ge 1$ , 1-strict pseudoconvexity at the boundary, \(\ell \) $\ell$ -uniform pseudoconvexity for \(\ell \ge 0\) $\ell \ge 0$ , and finally strong pseudoconvexity. Defining \(\ell \) $\ell$ -strict pseudoconvexity and \(\ell \) $\ell$ -uniform pseudoconvexity for \(\ell <2\) $\ell <2$ depends on extending a notion of strict plurisubharmonicity to cases lacking \(C^2\) $C^2$ -smoothness, first studying it in the sense of distribution and then considering it in infinite dimensions. Examples of strictly plurisubharmonic functions as well as strongly pseudoconvex domains related to important classical Banach spaces are presented. Finally, some related solutions to the inhomogeneous Cauchy–Riemann equations for \({\overline{\partial }}\) $\overline{\partial }$ -closed (0, 1)-forms in infinite-dimensional domains are shown.