Operator norm and lower bound of four-dimensional matrices

Motivated by the work “Hardy's inequality in several variables” in Salem et al. (2000), an extension of Copson's discrete inequality to multiple series is obtained. As an application of such extension, we consider the boundedness problem of four-dimensional matrices on the space Lp of double sequences. The operator norms of Cesàro, Copson and Nörlund four-dimensional matrices as operators selfmap of Lp are established. Further, a general upper estimate and lower estimate are obtained for the lower bounds of some four-dimensional matrices.