In this paper we introduce some new weighted maximal operators of the partial sums of the Walsh–Fourier series. We prove that for some “optimal” weights these new operators indeed are bounded from the martingale Hardy space $$H_{p}(G)$$ H p ( G ) to the Lebesgue space $$L_{p}(G),$$ L p ( G ) , for $$0<p<1.$$ 0 < p < 1 . Moreover, we also prove sharpness of this result. As a consequence we obtain some new and well-known results.

In this paper, we derive the convergence of 𝑇 means of Vilenkin–Fourier series with monotone coefficients of integrable functions in Lebesgue and Vilenkin–Lebesgue points. Moreover, we discuss the pointwise and norm convergence in L p L_{p} norms of such 𝑇 means.