  We develop local elliptic regularity for operators having coefficients in a
range of Sobolev-type function spaces (Bessel potential, Sobolev-Slobodeckij,
Triebel-Lizorkin, Besov) where the coefficients have a regularity structure
typical of operators in geometric analysis. The proofs rely on a nonstandard
technique using rescaling estimates and apply to operators having coefficients
with low regularity. For each class of function space for an operator's
coefficients, we exhibit a natural associated range of function spaces of the
same type for the domain of the operator and we provide regularity inference
along with interior estimates. Additionally, we present a unified set of
multiplication results for the function spaces we consider.
