Hello!

I'm reading the book of T. Aubin Some nonlinear problems in Riemannian geometry. In chapter 3 he introduces elliptic operators on manifolds, but then he gives regularity results for elliptic operators on bounded open sets of $\mathbb{R}^n$. How these results extend to the manifold setting? More precisely, let $(M,g)$ be a $n$-dimensional smooth compact Riemannian manifold without boundary, $T$ be a linear elliptic differential operator of order $2m$ with (in local coordinates) smooth coefficients.

Let $f\in L^{p}\left(M\right)$ with $1< p<\infty $ and suppose there exist $u\in L^{q}\left(M\right)$ with $1< q<\infty$ s.t.
$$\int_{M}u T^{*}\left(v\right)=\int_{M}fv$$
for all $v\in C^{\infty}\left(M\right)$ with $T^{*}$ the adjoint of $T$ w.r.t. the metric $g$. What can be said about $u$? Which kind of estimates does it satisfy? How can i use the theory on bounded open sets of euclidean spaces to get elliptic estimates on $u$?

EDIT: I'd like to prove the following proposition: Let $\left(M,g\right)$ compact Riemannian manifold without boundary, $T$ a linear elliptic operator of order $2m$ with smooth coefficients. Let $f\in L^{p}\left(M\right)$ with $1< p<\infty $ and suppose there exist $u\in L^{q}\left(M\right)$ with $1< q<\infty$ s.t. $$\int_{M}u T^{*}\left(v\right)=\int_{M}fv$$ for all $v\in C^{\infty}\left(M\right)$ with $T^{*}$ the adjoint of $T$ w.r.t. the metric $g$. Then $u\in W^{2m,p}\left(M\right)$ and satisfies the estimate

$$\left\|u\right\|_{W^{2m,p}\left(M\right)} \le \left(M,T,g,m,p\right)\left(\left\|f\right\|_{L^{p}\left(M\right)}+ \left\|u\right\|_{L^{p}\left(M\right)} \right)$$

Proof (tentative): I do the convolution $u_k = J_{\frac{1}{k}} \star u$ with $J_{\epsilon}$ the smoothing kernel. So i have

$$u_k\rightarrow u $$

in $L^{q}$ moreover for every $v\in C^{\infty}\left(M\right)$

$$\lim_{k\rightarrow +\infty}\int_{M}T\left(u_k\right)v=\lim_{k\rightarrow +\infty}\int_{M}u_k T^{*}\left(v\right)=\int_{M}u T^{*}\left(v\right)=\int_{M}f v $$

so $T\left(u_k\right)$ converges weakly in $L^p\left(M\right)$ to $f$ and because of this

$$\left\|T\left(u_k\right)\right\|_{L^{p}\left(M\right)}\leq C\left(f\right)$$

Now take an open covering $\lbrace B_{r,j}\rbrace_{j\in J}$ with $B_{r,j}$ open sets diffeomorphic to balls of radius $r$ sufficiently small and $\chi_j$ cutoff functions s.t. $\chi_{j}=1$ identically on $B_{2r,j}$ and $\chi_{j}=0$ identically outside $B_{3r,j}$. By Local theory i have that

$$\left\|\chi_j u_k\right\|_{W^{2m,p}\left(B_{3r,j}\right)}\leq C_j\left(\left\|T\left(\chi_j u_k\right)\right\|_{L^p\left(B_{3r,j}\right)}+ \left\|\chi_j u_k\right\|_{L^p\left(B_{3r,j}\right)}\right)\leq C_j\left(\left\|T\left(\chi_j u_k\right)\right\|_{L^p\left(B_{3r,j}\right)}+ \left\|u_k\right\|_{L^p\left(M\right)}\right)$$

Thanks to Deane i made some steps forward,

$$\left\|\chi_j u_k\right\|_{W^{2m,p}\left(B_{3r,j}\right)}\leq C_j\left(\left\|\chi_j T\left(u_k\right)\right\|_{L^p\left(B_{3r,j}\right)}+ \left\|u_k\right\|_{L^p\left(M\right)} + C_j\left(T\right)\left\|\chi_j \right\|_{W^{2m,\infty}\left(B_{3r,j}\right)}\left\|u_k\right\|_{W^{2m-1,p}\left(B_{3r,j}\right)}\right)$$

that becomes

$$\left\|\chi_j u_k\right\|_{W^{2m,p}\left(B_{3r,j}\right)}\leq C_j\left(\left\|T\left(u_k\right)\right\|_{L^p\left(M\right)}+ \left\|u_k\right\|_{L^p\left(M\right)} + C_j\left(T\right)\left\|\chi_j \right\|_{W^{2m,\infty}\left(B_{3r,j}\right)}\left\|u_k\right\|_{W^{2m-1,p}\left(B_{3r,j}\right)}\right)$$

By Ehrling's lemma (thm $7.30$ of Renardy-Rogers book Introduction to PDEs) i have that $\forall \epsilon_j>0$ exist $c\left(\epsilon_j\right)>0$ s.t.

$$\left\| u_k\right\|_{W^{2m-1,p}\left(B_{3r,j}\right)}\leq \epsilon_j\left\|u_k\right\|_{W^{2m,p}\left(B_{3r,j}\right)}+ c\left(\epsilon_j\right)\left\|u_k\right\|_{L^p\left(B_{3r,j}\right)}$$

that implies

$$\left\|\chi_j u_k\right\|_{W^{2m,p}\left(B_{3r,j}\right)}\leq \tilde{C}_j\left(T,\epsilon_j,\chi_j\right)\left[ \left\|T\left(u_k\right)\right\|_{L^p\left(M\right)}+ \left\|u_k\right\|_{L^p\left(M\right)} \right] + \epsilon_jC_j\left(T\right)\left\|\chi_j\right\|_{W^{2m,\infty}\left(B_{3r,j}\right)}\left\|u_k\right\|_{W^{2m,p}\left(M\right)}$$

Choosing $\epsilon_j$ small enough i have at last

$$\left\| u_k\right\|_{W^{2m,p}\left(M\right)}\leq \tilde{C}\left(T,\epsilon_j,\chi_j\right)\left[ \left\|T\left(u_k\right)\right\|_{L^p\left(M\right)}+ \left\|u_k\right\|_{L^p\left(M\right)} \right]$$

Now i suppose $p\geq q$ so

$$\left\| u_k\right\|_{W^{2m,p}\left(M\right)}\leq \tilde{C}\left(T,\epsilon_j,\chi_j\right)\left[ C\left(f\right)+ \left\|u\right\|_{L^q\left(M\right)} \right]$$

and by Sobolev embedding the best i can say is that $u\in W^{2m-1,p}\left(M\right)$ and it satisfies

$$\left\|u\right\|_{W^{2m-1,p}\left(M\right)}\leq C\left( C\left(f\right)+ \left\|u\right\|_{L^{q}\left(M\right)} \right)$$

How can i gain the $W^{2m,p}$ regularity and the estimates? How can i deal with the case $q$ is smaller than $p$?