ap.analysis of pdes – Reference request: \$L^p\$ regularity for elliptic differential operators

I am looking for a book that proves the following:

Let $$M$$ be a closed manifold, $$L$$ an elliptic differential operator on $$M$$ of order $$k$$ and with smooth coefficients. Suppose $$u$$ is a distribution on $$M$$ such that $$Luin L^p$$ ($$1), i.e., there exists $$vin L^p$$ such that $$langle u,L^*phirangle=int_Mv,phi$$ for all smooth $$phi$$, where $$L^*$$ is the formal adjoint of $$L$$, $$langlecdot,cdotrangle$$ is the dual pairing of distributions and test functions. Then $$uin W^{k,p}$$ and $$|u|_{W^{k,p}}leq C(|Lu|_{L^p}+|u|_{L^p})$$.

The case $$k=2$$ is discussed in many books on elliptic PDEs. Also, it seems that the case $$p=2$$ appears in many books, for example the first volume of M. E. Taylor’s trilogy Partial Differential Equations. However, I have not found a proof for the general case. Here are my questions:

1. Does this result necessarily rely on the theory of pseudodifferential operators?
2. What are some accessible books for learning its proof, with or without pseudodifferential operators? (For example, I believe this result is contained in Hörmander’s tetralogy on linear PDEs, but it could be very difficult to study from those books…)