REDUCTION THEOREMS FOR DISCRETE HARDY OPERATOR ON MONOTONE SEQUENCE CONES (0<p<1)

Abstract

In this paper, we consider the discrete Hardy and Copson operators on the cone of nonnegative monotone sequences. We prove that weighted inequalities of the form l_p→ l_q  for discrete Hardy and Copson operators on the cone of monotone sequences, in the case 0 < q < ∞, 0<p<1, can be reduced to the corresponding inequalities on the cone of nonnegative sequences. The latter possess a broader basis for proof, which significantly  extends the possibilities for their analysis. Weighted inequalities for the integral Hardy operator (in the continuous setting) on the cone of nonnegative nonincreasing functions have been studied previously by many authors. Reduction theorems for inequalities involving Hardy-type integral operators on the cone of nonincreasing functions to inequalities on the cone of nonnegative functions are also well known. We present various theorems concerning the equivalence of inequalities for discrete Hardy and Copson operators on the cone of nonnegative nonincreasing sequences and inequalities on the cone of nonnegative sequences. Our proofs differ substantially from those in the continuous case. Methods applicable in the continuous setting do not always work in the discrete setting. For the case p>1, analogous results were obtained by the authors earlier.