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Lifting of fractional Sobolev mappings to noncompact covering space (2301.07663v3)
Published 18 Jan 2023 in math.AP, math.CA, and math.FA
Abstract: Given compact Riemannian manifolds $\mathcal{M}$ and $\mathcal{N}$, a Riemannian covering $\pi : \smash{\widetilde{\mathcal{N}}} \to \mathcal{N}$ by a noncompact covering space $\smash{\widetilde{\mathcal{N}}}$, $1 < p < \infty$ and $0 < s < 1$, the space of liftings of fractional Sobolev maps in $\smash{\dot{W}{s, p}} (\mathcal{M}, \mathcal{N})$ is characterized when $sp > 1$ and an optimal nonlinear fractional Sobolev estimate is obtained when moreover $sp \ge \dim \mathcal{M}$. A nonlinear characterization of the sum of spaces $\smash{\dot{W}{s, p}} (\mathcal{M}, \mathbb{R}) + \smash{\dot{W}{1, sp}} (\mathcal{M}, \mathbb{R})$ is also provided.