L^p change of variables inequalities on manifolds

TL;DR

This paper establishes two-sided inequalities for the $L^p$-norms of pushforward and pullback operations on manifolds, generalizing classical change of variables results to differential forms using spectral properties.

Contribution

It introduces spectral-based bounds for differential forms on Riemannian manifolds, extending Jacobian-based inequalities from functions and densities.

Findings

Bounds depend only on Jacobian for functions and densities.

Spectral properties determine bounds for differential forms.

Generalizes change of variables inequalities to manifold settings.

Abstract

We prove two-sided inequalities for the $L^p$-norm of a pushforward or pullback (with respect to an orientation-preserving diffeomorphism) on oriented volume and Riemannian manifolds. For a function or density on a volume manifold, these bounds depend only on the Jacobian determinant, which arises through the change of variables theorem. For an arbitrary differential form on a Riemannian manifold, however, these bounds are shown to depend on more general "spectral" properties of the diffeomorphism, using an appropriately-defined notion of singular values. These spectral terms generalize the Jacobian determinant, which is recovered in the special cases of functions and densities (i.e., bottom and top forms).