In this paper we consider the problem of finding norms of certain matrix operators on sequence space bv p . In fact, we consider these problems for weighted mean, and generalized Cesaro matrices on sequence space bv p .Mathematics subject classification (2010): 26D15, 47A30, 40G05, 47D37, 46A45, 54D55.

In this paper, we obtained a necessary and sufficient condition for the embedding H?q([0,1])? IBVqp([0,1]), where IBVq p denotes the set of functions of bounded q-integral p-variation. Additionally, the conditions for the composition and superposition operators were provided to map the space H?q([0,1]) into itself, by which these operators were bounded. Finally, we applied these results to examine the existence and uniqueness of solutions to Hammerstein integral equations in the space of H?q([0,1]).

The aim of this paper is to establish the relation between the two classes of functions {\mathrm{BV}_{*}(q;\alpha)} and {\Lambda_{*}\mathrm{BV}^{p}}. It uses a new shorter proof and extends the results of Gogatishvili, Goginava and Tephnadze.