  In this paper we prove and discuss some new $\left( H_p,L_{p,\infty}\right)$
type inequalities of the maximal operators of $T$ means with monotone
coefficients with respect to Walsh-Kaczmarz system. It is also proved that
these results are the best possible in a special sense. As applications, both
some well-known and new results are pointed out. In particular, we apply these
results to prove a.e. convergence of such $T$ means.
