Regularity of functions smooth along foliations, and elliptic regularity

We produce two sets of results arising in the analysis of the degree of smoothness of a function that is known to be smooth along the leaves of one or more foliations. These foliations might arise from Anosov systems, and while each leaf is smooth, the leaves might vary in a nonsmooth fashion. One set of results gives microlocal regularity of such a function away from the conormal bundle of a foliation. The other set of results gives local regularity of solutions to a class of elliptic systems with fairly rough coefficients. Such a regularity theory is motivated by one attack on the foliation regularity problem. © 2005 Elsevier Inc. All rights reserved.