In Ho and Russell (SIAM J Control Optim 21(4): 1983), and Weiss (Syst Control Lett 10(1): 79-82, 1988), a Carleson measure criterion for admissibility of one-dimensional input elements with respect to diagonal semigroups is given. We extend their results from the Hilbert space situation (L 2 -admissibility on the state space 2 ) to the more general situation of L p -admissibility on the state space q . For analytic diagonal semigroups we present a new result that does not rely on Laplace transform methods. A comparison of both criteria leads to a result on L p -admissibility for reciprocal systems in the sense of Curtain (Syst Control Lett 49(2): [81][82][83][84][85][86][87][88][89] 2003).