Best possible results in a class of inequalities

Peter Dexter Johnson, Jr. and R. N. Mohapatra

Vol. 103 (1982), No. 2, 433–436

DOI: 10.2140/pjm.1982.103.433

Abstract

In this paper we shall prove the following theorem.

Theorem. Suppose 1 < p ≦∞, and rp > 1 if p < ∞, r > 0 if p = ∞. Suppose the matrix A = (a_nk) with a_nk = n^−r (k ≦ n), a_nk = 0 (k > n). Suppose w be the subset of w consisting of nonnegative, monotone sequences. Then {n^r−1}_n is maximum, with respect to <, in I where

I = {b ∈w :  for some K > 0,∥A|bx|∥p ≦ K∥x∥p for all x ∈ lp}.

Mathematical Subject Classification 2000

Primary: 26D15

Secondary: 40C05

Milestones

Received: 28 July 1980

Published: 1 December 1982