Sobolev smoothness and invertibility conjecture for the operator $\Psi^L$

Let $M$ be a $d$-dimensional manifold, let $N$ be a Riemannian manifold, and let $\operatorname{Imm}^k(M,N)$ denote the Sobolev completion of the space of immersions. For an immersion $f$, write $\Gamma_{H^s}(f^*TN)$ for the Sobolev sections of $f^*TN$ of order $s$, and let $\Psi^L$ be the operator associated with the metric determined by $L$.

Smoothness and invertibility conjecture. For each $k>\frac{d}{2}+1$, the operator $\Psi^L$ depends smoothly on the immersion $f\in\operatorname{Imm}^k(M,N)$ and is invertible as a mapping

$$\Psi^L:\Gamma_{H^{k+2-2l}}(f^*TN)\longrightarrow\Gamma_{H^k}(f^*TN).$$

This result is intended to extend the projection construction to smooth mappings on Sobolev completions. The required elliptic theory for pseudodifferential operators with Sobolev coefficients is stated as a missing ingredient and is deferred to future work, so the claim is currently open.