Higher complex Sobolev spaces on complex manifolds

TL;DR

This paper explores higher complex Sobolev spaces on complex manifolds, establishing inequalities and connections to complex Monge-Ampère equations, advancing understanding of their analytical and geometric properties.

Contribution

It introduces and analyzes higher complex Sobolev spaces, proves the Moser-Trudinger inequality for these spaces, and explores their relationship with the complex Monge-Ampère equation.

Findings

Proved the Moser-Trudinger inequality for higher complex Sobolev spaces.

Established links between these Sobolev spaces and the complex Monge-Ampère equation.

Enhanced understanding of the functional and geometric properties of these spaces.

Abstract

We study higher complex Sobolev spaces and their corresponding functional capacities. In particular, we prove the Moser-Trudinger inequality for these spaces and discuss some relationships between these spaces and the complex Monge-Amp\`{e}re equation.