Random Clarkson inequalities and LP versions of Grothendieck's inequality

Abstract

AbstractIn a recent paper KATO [3] used the LITTLEWOOD matrices to generalise CLARKSON'S inequalities. Our first aim is to indicate how KATO'S result can be deduced from a neglected version of the HAUSDORFF‐YOUNG inequality which was proved by WELLS and WILLIAMS [12]. We next establish „random CLARKSON inequalities”︁. These show that the expected behaviour of matrices whose coefficients are random ± 1′s is, as one might expect, the same as the behaviour that KATO observed in the LITTLEWOOD matrices. Finally we show how sharp Lp versions of GROTHENDIECK'S inequality can be obtained by combining a KATO‐like result with a theorem of BENNETT [1] on SCHUR multipliers.