On the Continuity of Exterior Differentiation Between Sobolev-Slobodeckij Spaces of Sections of Tensor Bundles on Compact Manifolds

Ali Behzadan
UCSD

Abstract:

Suppose Ω is a nonempty open set with Lipschitz continuous boundary in \mathbbR^{n}. There are certain exponents
e ∈ R and q ∈ (1,∞) for which [(∂)/(∂x^{j})]: W^{e,q}(Ω)→ W^{e−1,q}(Ω) is NOT a well-defined continuous operator. Now suppose M is a compact smooth manifold. In this talk we will try to discuss the
following questions:

1. How are Sobolev spaces of sections of vector bundles on M defined?
2. Is it possible to extend d: C^{∞}(M)→ C^{∞}(T^{*}M) to a continuous linear map from W^{e,q}(M) to W^{e−1,q}(T^{*}M) for all e ∈ R and q ∈ (1,∞)?
3. Why are we interested in the above question?