The One-dimensional Martingale Hardy Spaces and Partial Sums and Fej\'er Means with respect to Walsh system

In this paper we prove and discuss some new $\left( H_p,L_{p}\right)$ type inequalities for partial Sums and Fejer means with respect to Walsh 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.