Abstract. The purpose of this paper is to obtain some new estimates of the norms of functions in Lorentz spaces $L_{q,r}$ ($q\ge2,r>0$), which are analogous to the estimates given by V.I. Kolyada [3] in the case of Lebesgue spaces. We shall also get estimates that generalize a well-known theorem of Marcinkiewicz and Zygmund [6]. The second half of our paper is devoted to obtain conjugate estimates for the ones mentioned above.

AMS Subject Classification (1991): 46E35, 26A15

Received May 1, 1998 and in revised form October 2, 1998. (Registered under 2681/2009.)