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Weighted inequalities for sub-monotone functionals

Amiran Gogatishvili, Luboš Pick

TL;DR

The paper addresses unifying weighted inequalities for various integral operators through the notion of sub-monotone functionals. It introduces a universal equivalence framework: for $p$ in $(1,\infty)$ and a weight $v$, the basic inequality $\varrho\left(\tfrac{1}{V(t)}\int_{0}^{t} f(s) v(s) ds\right) \le C (\int_{0}^{\infty} f(t)^p v(t) dt)^{1/p}$ is shown to be equivalent to a broad spectrum of transformed inequalities (including Hardy, Copson, geometric mean, and harmonic mean forms) labeled (ii)–(ix). This unifies disparate classical results—such as those of Sinnamon, Maz’ya–Rozin, and Sinnamon–Stepanov—via elementary transformations, revealing deep interconnections between nonlinear operators and Hardy-type bounds. The proofs rely on elementary tools (Jensen, Hölder, and lattice properties) to establish a cobweb of implications that collectively reduce complex nonlinear inequalities to the classical Hardy framework. The resulting framework offers a versatile method to derive new inequalities by transferring known results across operators and parameter regimes, with potential extensions to cones of monotone functions and related operator families.

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.

Weighted inequalities for sub-monotone functionals

TL;DR

The paper addresses unifying weighted inequalities for various integral operators through the notion of sub-monotone functionals. It introduces a universal equivalence framework: for in and a weight , the basic inequality is shown to be equivalent to a broad spectrum of transformed inequalities (including Hardy, Copson, geometric mean, and harmonic mean forms) labeled (ii)–(ix). This unifies disparate classical results—such as those of Sinnamon, Maz’ya–Rozin, and Sinnamon–Stepanov—via elementary transformations, revealing deep interconnections between nonlinear operators and Hardy-type bounds. The proofs rely on elementary tools (Jensen, Hölder, and lattice properties) to establish a cobweb of implications that collectively reduce complex nonlinear inequalities to the classical Hardy framework. The resulting framework offers a versatile method to derive new inequalities by transferring known results across operators and parameter regimes, with potential extensions to cones of monotone functions and related operator families.

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.
Paper Structure (2 sections, 4 theorems, 107 equations)

This paper contains 2 sections, 4 theorems, 107 equations.

Table of Contents

  1. Introduction and the main results
  2. Proofs

Key Result

Theorem 1.1

Assume that $\varrho\colon \mathcal{M}_+(0,\infty)\to [0,\infty]$ is a sub-monotone functional. Let $p\in(1,\infty)$ and let $v$ be a weight. Then the following statements are equivalent: (i) there exists a positive constant $C_1$ such that for every $f\in\mathcal{M}_+(0,\infty)$, (ii) for every $r\in[1,\infty)$ and $\alpha\in (\max\{-\frac{1}{p}, -\frac{1}{p'}\},\infty)$ there exist positive con

Theorems & Definitions (7)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1
  • Remark 2.2