Abstract
Over the (1, N)-dimensional supercircle S-1 vertical bar N, we classify n(1 vertical bar N)-invariant linear differential operators acting on the superspaces of weighted densities on S-1 vertical bar N, where n(1 vertical bar N) is the Heisenberg Lie superalgebra. This result allows us to compute the first differential n(1 vertical bar N)-relative cohomology of the Lie superalgebra K(N) of contact vector fields with coefficients in the superspace of weighted densities. For N = 0, 1, 2, we investigate the first n(1 vertical bar N)-relative cohomology space associated with the embedding of K(N) in the superspace of the supercommutative algebra SP(N) of pseudodifferential symbols on S-1 vertical bar N and in the Lie superalgebra S Psi DO(S-1 vertical bar N) of superpseudodifferential operators with smooth coeffcients. We explicity give 1-cocycles spanning these cohomology spaces.