We discuss (Hpβ,Lpβ) and (Hpβ,weakβLpβ) type inequalities of
weighted maximal operators of T means with respect to the Vilenkin systems
with monotone coefficients, considered in \cite{tut4} and prove that these
results are the best possible in a special sense. As applications, both some
well-known and new results are pointed out.Comment: arXiv admin note: substantial text overlap with arXiv:2101.09196,
arXiv:1504.05974 by other author