Goginava proved that the maximal operator σ α , * {\sigma^{\alpha,*}} ( 0 < α < 1 {0<\alpha<1}) of two-dimensional Marcinkiewicz type (C , α) {(C,\alpha)} means is bounded from the two-dimensional dyadic martingale Hardy space H p (G 2) {H_{p}(G^{2})} to the space L p (G 2) {L^{p}(G^{2})} for p > 2 2 + α {p>\frac{2}{2+\alpha}}. Moreover, he showed that assumption p > 2 2 + α {p>\frac{2}{2+\alpha}} is essential for the boundedness of the maximal operator σ α , * {\sigma^{\alpha,*}}. It was shown that at the point p 0 = 2 2 + α {p_{0}=\frac{2}{2+\alpha}} the maximal operator σ α , * {\sigma^{\alpha,*}} is bounded from the dyadic Hardy space H 2 / (2 + α) (G 2) {H_{2/(2+\alpha)}(G^{2})} to the space weak- L 2 / (2 + α) (G 2) {L^{2/(2+\alpha)}(G^{2})}. The main aim of this paper is to investigate the behaviour of the maximal operators of weighted Marcinkiewicz type σ α , * {\sigma^{\alpha,*}} means ( 0 < α < 1 {0<\alpha<1}) in the endpoint case p 0 = 2 2 + α {p_{0}=\frac{2}{2+\alpha}}. In particular, the optimal condition on the weights is given which provides the boundedness from H 2 / (2 + α) (G 2) {H_{2/(2+\alpha)}(G^{2})} to L 2 / (2 + α) (G 2) {L^{2/(2+\alpha)}(G^{2})}. Furthermore, a strong summation theorem is stated for functions in the dyadic martingale Hardy space H 2 / (2 + α) (G 2) {H_{2/(2+\alpha)}(G^{2})}. [ABSTRACT FROM AUTHOR]