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In this paper we prove and discuss some new $\left( H_p,L_{p}\right)$ typeinequalities for partial Sums and Fej\'er 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.

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