UPPER ESTIMATE FOR THE NORM OF HAUSDORFF MATRICES ON THE SEQUENTIAL WEAK lp SPACE

Let A = (an;k)n;k¸0 be a non-negative matrix. Denote by kAklp;weak lp the infimum of those U; satisfying the following inequality: sup B ( 1 (#B)1¡1 p X n2B Ã 1X k=0 an;kxk !) · U Ã 1X k=0 xp k !1=p ; where x = fxkg1k =0 2 lp; xk ¸ 0 and the sup is taken over all nonempty subset B ½ N with finite cardinal. In this paper we focus on the evaluation of kAklp;weak lp , where A is a Hausdorff matrix operator. A Hardy type formula is established as an upper estimate. In particular, we apply our results to Ces`aro matrices, H¨older matrices, Euler matrices and Gamma matrices.