Abstract. We establish some general theorems for a wide class of sequences of superlinear operators $\{T_n : L^1(I^2) \to L^0(I^2),n = 1,2,\ldots\} $ about existence of a function $g$ from a certain class $L\phi(L)$ such that the sequence of functions $\{T_{n}(g), n=1,2,\ldots\} $ is not bounded in measure. These theorems imply several results about divergence for classical operators with respect to general and classical orthonormal systems.

AMS Subject Classification (1991): 42C15, 42C10

Received April 2, 2004, and in revised form December 15, 2004. (Registered under 5866/2009.)