We establish appropriate $L^p$ estimates for a class of maximal operators $S_{Omega}^{(gamma)}$ on the product space $R^n; imes;R^m;when;Omega$ lacks regularity and $1;le;gamma;le;2.;Also,;when;gamma;=;2$, we prove the $L^p;(2;{le};P;<;infty);boundedness;of;S_{Omega}^{(gamma)};whenever;Omega$ is a function in a certain block space $B_q^{(0,0)}(S^{n-1}; imes;S^{m-1})$ (for some q > 1). Moreover, we show that the condition $Omega;{in};B_q^{(0,0)}(S^{n-1}; imes;S^{m-1})$ is nearly optimal in the sense that the operator $S_{Omega}^{(2)}$ may fail to be bounded on $L^2$ if the condition $Omega;{in};B_q^{(0,0)}(S^{n-1}; imes;S^{m-1})$ is replaced by the weaker conditions $Omega;{in};B_q^{(0,varepsilon)}(S^{n-1}; imes;S^{m-1});for;any;-1;<;varepsilon;<;0.$