WEIGHTED INEQUALITIES FOR A SUPERPOSITION OF THE THREE OPERATORS

We study a three-weight inequality for a superposition of the Copson, Hardy, and Tandori operators. The goal of this paper is to prove a complete characterization of the boundedness of the operator that is a combination of these three operators in weighted Lebesgue spaces from  to . The main focus is on determining necessary and sufficient conditions under which this inequality holds for all non-negative measurable functions on the positive real axis. The notion of a fundamental function of a Borel measure with respect to an increasing function is used substantially. Since the Tandori operator is not a linear operator, we cannot use the duality methods used in earlier works. To solve this problem, we develop a new, simplified discretization method that avoids the complexities of previously known methods. An explicit form of the best constant in the inequality is obtained, demonstrating the accuracy and optimality of the results. By establishing necessary and sufficient conditions for the boundedness of these composite operators, we improve the inequalities previously established in the works of Gogatishvili A., Pick L., Opic B. [1]. The results obtained in the paper extend and complement existing research in the field of weighted inequalities and operator analysis in function spaces and offer potential applications in approximation theory, harmonic analysis and related areas.

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