Rough metrics on manifolds and quadratic estimates

Abstract

Abstract. We study the persistence of quadratic estimates related to the Kato square root problem across a change of metric on smooth manifolds by defining a class of Riemannian-like metrics that are permitted to be of low regularity and degenerate on sets of measure zero. We also demonstrate how to transmit quadratic estimates between manifolds which are homeomorphic and locally bi-Lipschitz. As a consequence, we demonstrate the invariance of the Kato square root problem under Lipschitz transformations of the space and obtain solutions to this problem on functions and forms on compact manifolds with a continuous metric. Furthermore, we show that a lower bound on the injectivity radius is not a necessary condition to solve the Kato square root problem.

Self-adjointness of the Gaffney Laplacian on Vector Bundles

“…Now assume that s is essentially self-adjoint. Taking closures in (L) and using Proposition 3.4 we obtain (3), which leads to D = s = N , that is, W 1,2 0 (V) = W 1,2 (V). Now by Theorem 5.1 and Proposition 3.5, it follows that (M, V, ∇) has negligible boundary.…”

“…Some initial progress in this direction can be found in [3] for the special case of Sobolev spaces of functions under so-called "rough metrics." These considerations are beyond the scope of this paper and we will always assume compatibility between the metric and connection unless otherwise stated.…”

Essential self-adjointness of powers of first-order differential operators on non-compact manifolds with low-regularity metrics

“…Computing in L 2 allows us to assert that div g f = θ −1 divg(θEf ) (c.f. Proposition 12 in [4]). Therefore,…”

“…This allows us to define measurable functions f : M → C, and by Γ(V), we denote measurable sections over V to be sections v = v i e i in continuous local frames {e i } with v i a measurable function. See [4] for a detailed construction.…”

“…Proof. Since the metric g is at least continuous, we note as in Proposition 13 of [4] that, given any C > 1, there is a smooth metricg which is C-close to g, by which we mean that: C −1 |u|g ≤ |u| g ≤ C |u|g for every x ∈ M and u ∈ T x M.…”