  The main results of our paper deal with the lifting problem for multilinear
differential operators between complexes of horizontal de Rham forms on the
infinite jet bundle. We answer the question when does an n-multilinear
differential operator from the space of (N,0)-forms (where N is the dimension
of the base) to the space of (N-s,0)-forms allow an n-multilinear extension of
degree (-s,0) defined on the whole horizontal de Rham complex.
  To study this problem we define a differential graded operad DEnd of
multilinear differential endomorphisms, which we prove to be acyclic in
positive degrees (negative mapping degrees) and describe the cohomology group
in degree zero in terms of the characteristic.
  As a corollary, we solve the lifting problem. An important application to
mathematical physics is the proof of existence of a strongly homotopy Lie
algebra structure extending a Lie bracket on the space of functionals.
