Over the (1,N)-dimensional supercircle S1∣N, we classify n(1∣N)-invariant linear differential operators acting on the superspaces of weighted densities on S1∣N, where n(1∣N) is the Heisenberg Lie superalgebra. This result allows us to compute the first differential n(1∣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∣N)-relative cohomology space associated with the embedding of K(N) in the superspace of the supercommutative algebra SP(N) of pseudodifferential symbols on S1∣N and in the Lie superalgebra SΨDO(S1∣N) of superpseudodifferential operators with smooth coeffcients. We explicity give 1-cocycles spanning these cohomology spaces