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A Sharp Localized Weighted Inequality Related to Gagliardo and Sobolev Seminorms and Its Applications

Pingxu Hu, Yinqin Li, Dachun Yang, Wen Yuan

Abstract

In this article, we establish a nearly sharp localized weighted inequality related to Gagliardo and Sobolev seminorms, respectively, with the sharp $A_1$-weight constant or with the specific $A_p$-weight constant when $p\in (1,\infty)$. As applications, we further obtain a new characterization of Muckenhoupt weights and, in the framework of ball Banach function spaces, an inequality related to Gagliardo and Sobolev seminorms on cubes, a Gagliardo--Nirenberg interpolation inequality, and a Bourgain--Brezis--Mironescu formula. All these obtained results have wide generality and are proved to be (nearly) sharp. The original version of this article was published in [Adv. Math. 481 (2025), Paper No. 110537]. In this revised version, we correct an error appeared in Theorem 1.1 in the case where $p=1$, which was pointed out to us by Emiel Lorist.

A Sharp Localized Weighted Inequality Related to Gagliardo and Sobolev Seminorms and Its Applications

Abstract

In this article, we establish a nearly sharp localized weighted inequality related to Gagliardo and Sobolev seminorms, respectively, with the sharp -weight constant or with the specific -weight constant when . As applications, we further obtain a new characterization of Muckenhoupt weights and, in the framework of ball Banach function spaces, an inequality related to Gagliardo and Sobolev seminorms on cubes, a Gagliardo--Nirenberg interpolation inequality, and a Bourgain--Brezis--Mironescu formula. All these obtained results have wide generality and are proved to be (nearly) sharp. The original version of this article was published in [Adv. Math. 481 (2025), Paper No. 110537]. In this revised version, we correct an error appeared in Theorem 1.1 in the case where , which was pointed out to us by Emiel Lorist.
Paper Structure (18 sections, 37 theorems, 164 equations)

This paper contains 18 sections, 37 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. Proofs of Theorems \ref{['thm-upi']} and \ref{['thm-ap']}
  3. An Inequality Related to Gagliardo and Sobolev Seminorms in Framework of Ball Banach Function Spaces $X(Q)$ over Cubes
  4. Ball Banach Function Spaces and Extrapolation
  5. An Inequality Related to Gagliardo and Sobolev Seminorms in Framework of $X(Q)$
  6. A Fractional Gagliardo--Nirenberg Inequality in Framework of Ball Banach Function Spaces
  7. A BBM Formula and A New Characterization of Ball Banach Sobolev Spaces
  8. A BBM Formula on Ball Banach Function Spaces
  9. A New Characterization of Ball Banach Sobolev Spaces
  10. The Sharpness of The Triebel--Lizorkin Exponent $q$
  11. Applications to Specific Function Spaces
  12. Bourgain--Morrey-Type Spaces
  13. Mixed-Norm Lebesgue Spaces
  14. Variable Lebesgue Spaces
  15. Lorentz Spaces
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $k\in \mathbb{N}$ and $p\in[1,\infty)$.

Theorems & Definitions (97)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 87 more