Weighted inequalities for sub-monotone functionals

TL;DR

This paper develops a general framework for weighted inequalities involving sub-monotone functionals and integral operators, enabling problem transfer and including nonlinear operators like geometric and harmonic means.

Contribution

It introduces a broad, novel set of relations for weighted inequalities with sub-monotone functionals, applicable to diverse integral and nonlinear operators.

Findings

Established relations between weighted inequalities and sub-monotone functionals.

Enabled problem transfer for solving complex inequalities.

Included nonlinear operators such as geometric and harmonic means.

Abstract

We establish a set of relations between several quite diverse types of weighted inequalities involving various integral operators and fairly general quasinorm-like functionals which we call sub-monotone. The main result enables one to solve a specific problem by transferring it to another one for which a solution is known. The main result is formulated in a rather surprising generality, involving previously unknown cases, and it works even for some nonlinear operators such as the geometric or harmonic mean operators. Proofs use only elementary means.