Fine properties of functions in complex Sobolev spaces

TL;DR

This paper investigates local properties of functions in complex Sobolev spaces, introducing a new capacity concept, establishing inequalities, and extending key results in pluripotential theory.

Contribution

It introduces a novel functional capacity for complex Sobolev spaces, proves sharp inequalities, and strengthens existing results in pluripotential theory.

Findings

Established a sharp inequality between the new capacity and Bedford-Taylor capacity.

Proved that the functional capacity is a Choquet capacity.

Characterized the Moser-Trudinger inequality via volume-capacity inequality.

Abstract

We study comprehensively local properties of functions in complex Sobolev spaces on a bounded open subset of $\mathbb{C}^n$. The main tool is the corresponding functional capacity for the space which is inspired by the global one due to Vigny (2007). An inequality between this capacity and the Bedford-Taylor capacity for plurisubharmonic functions is proved, which is sharp as far as the exponents are concerned. Moreover, it is shown that the functional capacity is a Choquet capacity. The Alexander-Taylor type inequality for the capacity is also proved. This allows us to strengthen the results in the works of Dinh, Marinescu and Vu (2023), Vigny and Vu (2024). Lastly, the Moser-Trudinger type inequality in this space is characterized by the volume-capacity inequality.