L^p change of variables inequalities on manifolds

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).

13 pages; v2: reformatting and some very minor revisions, as accepted for publication

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