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Some reverse inequalities for scalar Birkhoff weak integrable functions

Anca Croitoru, Alina Iosif, Anna Rita Sambucini, Luca Zampogni

TL;DR

The paper addresses the challenge of extending classical Hölder and Minkowski inequalities to the Birkhoff weak integral with non-additive measures, establishing reverse forms in the sublinear regime $0<p<1$ and exploring broader contexts. The authors develop reverse Hölder and reverse Minkowski inequalities for scalar $B_w$-integrable functions, derive weighted Hölder-type relations, and handle edge cases, then extend these results to vector-valued and interval-valued settings. They further demonstrate applications in interval analysis and propose a sparsity-promoting reconstruction model within the quasi-Banach framework, highlighting the practical relevance to uncertainty modeling and robust signal processing. The work broadens the scope of generalized $L^p$-type spaces under non-additive integration and provides tools for applications in interval analysis, fuzzy control, and image reconstruction.

Abstract

Some inequalities and reverses of classic Hölder and Minkowski types are obtained for scalar Birkhoff weak integrable functions with respect to a non-additive measure.

Some reverse inequalities for scalar Birkhoff weak integrable functions

TL;DR

The paper addresses the challenge of extending classical Hölder and Minkowski inequalities to the Birkhoff weak integral with non-additive measures, establishing reverse forms in the sublinear regime and exploring broader contexts. The authors develop reverse Hölder and reverse Minkowski inequalities for scalar -integrable functions, derive weighted Hölder-type relations, and handle edge cases, then extend these results to vector-valued and interval-valued settings. They further demonstrate applications in interval analysis and propose a sparsity-promoting reconstruction model within the quasi-Banach framework, highlighting the practical relevance to uncertainty modeling and robust signal processing. The work broadens the scope of generalized -type spaces under non-additive integration and provides tools for applications in interval analysis, fuzzy control, and image reconstruction.

Abstract

Some inequalities and reverses of classic Hölder and Minkowski types are obtained for scalar Birkhoff weak integrable functions with respect to a non-additive measure.

Paper Structure

This paper contains 10 sections, 13 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Birkhoff weak integrability and related inequalities
  4. Applications
  5. Vector valued case
  6. Applications in Interval Analysis
  7. Interval-valued functions and scalar set functions ($F,\nu$)
  8. Scalar functions and interval-valued set functions ($u,M$)
  9. A sparsity-promoting reconstruction model
  10. Conclusion

Key Result

Theorem 3.2

(CG) Let $u, v:T\rightarrow \mathbb{R}$, such that $u, v\in B_w(\nu)$. Then $\min\{u, v\}$ and $\max\{u, v\}$ are in $B_w(\nu).$

Theorems & Definitions (38)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 3.1
  • Theorem 3.2
  • Definition 3.3
  • Example 3.4
  • Theorem 3.5
  • Proposition 3.6
  • ...and 28 more