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 $Luin L^p$ ($1<p<infty$), 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…)