Projectively Invariant Hardy Spaces on Domains with Corners

TL;DR

This paper develops projectively invariant Hardy spaces on domains with corners by constructing invariant measures on singular boundary parts, extending classical smooth domain theory to piecewise smooth domains with corners.

Contribution

It introduces a new projectively invariant measure on singular boundary parts of domains with corners, linking positivity to strong $ ext{C}$-convexity and Hardy space construction.

Findings

Invariant measure on singular boundary parts constructed

Positivity of the measure aligns with strong $ ext{C}$-convexity

Hardy spaces extended to domains with corners

Abstract

For smoothly bounded, strongly $\mathbb{C}$-convex domains, one can use the Fefferman form or its variants to define projectively invariant norms on sections of holomorphic line bundles, producing a Hardy space. In two variables, we construct a projectively invariant measure on the singular part of a piece-wise smooth domain, and show that positivity of this invariant coincides with a notion of strong $\mathbb{C}$-convexity that is compatible with Cauchy-Fantappi\'e-Leray kernels, and thus define projectively invariant Hardy spaces as in the smooth case.