A Note on Hardy Spaces on Quadratic CR Manifolds

TL;DR

This paper investigates Hardy spaces on quadratic CR manifolds, demonstrating that the $L^p$-norms of holomorphic functions restricted to translates of the manifold decrease according to a specific convex ordering related to the Levi form.

Contribution

It introduces a new monotonicity property of $L^p$-norms for holomorphic functions on quadratic CR manifolds, connecting geometric properties to function space behavior.

Findings

$L^p$-norms decrease along certain translations of the manifold

The decrease is governed by the convex envelope of the Levi form image

Provides insights into Hardy space behavior on CR manifolds

Abstract

Given a quadratic CR manifold $\mathcal{M}$ embedded in a complex space, and a holomorphic function $f$ on a tubular neighbourhood of $\mathcal{M}$, we show that the $L^p$-norms of the restriction of $f$ to the translates of $\mathcal{M}$ is decreasing for the ordering induced by the closed convex envelope of the image of the Levi form of $\mathcal{M}$.