Convexity, plurisubharmonicity and the strong maximum modulus principle in Banach spaces

TL;DR

This paper explores the analogy between convexity and plurisubharmonicity in Banach spaces, introduces a strict version of the latter, and applies it to analyze the strong maximum modulus principle in $L^p$ direct integrals, providing a new, unified proof.

Contribution

It introduces a notion of strict plurisubharmonicity in Banach spaces and applies it to establish a concise, unified proof of the strong maximum modulus principle for $L^p$ direct integrals.

Findings

$L^p$ direct integrals satisfy the strong maximum modulus principle if and only if almost all components do.

The notion of strict plurisubharmonicity offers a new proof technique for the principle.

The approach unifies and simplifies existing results in the theory.

Abstract

In this article, we first try to make the known analogy between convexity and plurisubharmonicity more precise. Then we introduce a notion of strict plurisubharmonicity analogous to strict convexity, and we show how this notion can be used to study the strong maximum modulus principle in Banach spaces. As an application, we define a notion of $L^p$ direct integral of a family of Banach spaces, which includes at once Bochner $L^p$ spaces, $\ell^p$ direct sums and Hilbert direct integrals, and we show that under suitable hypotheses, when $p < \infty$, an $L^p$ direct integral satisfies the strong maximum modulus principle if and only if almost all members of the family do. This statement can be considered as a rewording of several known results, but the notion of strict plurisubharmonicity yields a new proof of it, which has the advantage of being short, enlightening and unified.