REDUCTION THEOREMS FOR DISCRETE HARDY OPERATOR ON MONOTONE SEQUENCE CONES (0 < p < 1)
Монотонды тiзбектер конусындағы дискреттi Харди операторы үшiн редукциялық теоремалар (0 < p < 1)
Редукционные теоремы для дискретного оператора Харди на конусах монотонных последовательностей (0 < p < 1)
Bokayev N. Gogatishvili A. Kuzeubayeva N.
25 December 2025al-Farabi Kazakh State National University
KazNU Bulletin. Mathematics, Mechanics, Computer Science Series
2025#128Issue 412 - 24 pp.
In this work, we investigate the discrete Hardy and Copson operators acting on the cone of nonnegative monotone sequences. It is established that the weighted inequalities of type lp → lq for these operators, in the case 0 < q < ∞, 0 < p < 1, can be reduced to the corresponding inequalities defined on the cone of general 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 well established. In this paper, we provide several theorems that demonstrate the equivalence between inequalities for discrete Hardy and Copson operators on the cone of nonnegative nonincreasing sequences and the corresponding 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.
Copson operator , discrete Hardy operator , monotone sequences , reduction theorems , weighted inequalities
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Department of Fundamental Mathematics at L.N. Gumilyov Eurasian National University, Astana, Kazakhstan
Institute of Mathematics of the Czech Academy of Sciences, Praha, Czech Republic
Department of Fundamental Mathematics at L.N. Gumilyov Eurasian National University
Institute of Mathematics of the Czech Academy of Sciences
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