We prove that if A : D(A) ⊂ L p (Ω; V ) → L p (Ω; W ) is a k th order constant-rank differential operator with maximal rank, then there exists a linear solution operator A −1 satisfying Sobolev regularity estimates. This allows us to construct a linear projection T :for all sufficiently regular maps. The estimate generalizes Fuch's distance estimate for the del-var operator, to operators such as the laplacian and the divergence. On regular domains, the same estimates are (trivially) observed for operators with finite dimensional null-space. When A is only assumed to be of constant rank, we are able to show that the Sobolev distance to the space of A-harmonic maps is bounded by Au, that is, minSimilar projection and distance estimates involving lower-order derivatives are further discussed, as well as a plethora of examples and applications with well-known operators. Contents 1. Introduction and main results 1 2. Set-up and preliminary results 10 3. Proofs of the main results 15 4. Examples of maximal-rank operators and applications 19 References 23