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On general Caffarelli-Kohn-Nirenberg type inequalities involving non-doubling weights in the case of $p=1$

Toshio Horiuchi

TL;DR

This work extends Caffarelli-Kohn-Nirenberg type inequalities to the case p=1 with general, including non-doubling, weights by introducing the weight framework $W(\mathbb{R}_+)$, their monotone rearrangements ($\varphi_w$, $\psi_w$) and the derived $v_w$, $V_w^q$ functionals. A non-degenerate condition (NDC) on $K(r)=|w(r)/(rw'(r))|$ near the origin governs the validity of the inequalities, enabling a robust $n$-dimensional CKN-type inequality via radial reduction and a careful change of variables. The paper proves a 1D inequality with optimal constants, lifts it to higher dimensions under NDC, and characterizes both necessity (via vanishing or explode-at-origin weights) and insufficiency (when $K(r)\to0$) regimes. An appendix consolidates non-critical CKN-type inequalities and symmetry properties of the best constants. Overall, the results provide a unified framework for weighted CKN inequalities at p=1 with broad weight classes and explicit constant behavior.

Abstract

We study the Caffarelli-Kohn-Nirenberg type inequalities in the case of $p=1$ and generalize them adopting weight functions $w(|x|)$ on $R^n$ with $w(t)$ in ${W}(R_+)$. Here ${W}(R_+)$ is a general class of weight functions on $R_+$ including non-doubling weights like $e^{1/t}$ and $e^{-1/t}$.

On general Caffarelli-Kohn-Nirenberg type inequalities involving non-doubling weights in the case of $p=1$

TL;DR

This work extends Caffarelli-Kohn-Nirenberg type inequalities to the case p=1 with general, including non-doubling, weights by introducing the weight framework , their monotone rearrangements (, ) and the derived , functionals. A non-degenerate condition (NDC) on near the origin governs the validity of the inequalities, enabling a robust -dimensional CKN-type inequality via radial reduction and a careful change of variables. The paper proves a 1D inequality with optimal constants, lifts it to higher dimensions under NDC, and characterizes both necessity (via vanishing or explode-at-origin weights) and insufficiency (when ) regimes. An appendix consolidates non-critical CKN-type inequalities and symmetry properties of the best constants. Overall, the results provide a unified framework for weighted CKN inequalities at p=1 with broad weight classes and explicit constant behavior.

Abstract

We study the Caffarelli-Kohn-Nirenberg type inequalities in the case of and generalize them adopting weight functions on with in . Here is a general class of weight functions on including non-doubling weights like and .
Paper Structure (7 sections, 7 theorems, 63 equations)

This paper contains 7 sections, 7 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Proof of Theorem \ref{['Thm4.1']}
  4. Proof of Theorem \ref{['thm4.2']}
  5. Proof of Theorem \ref{['Thm4.3']}
  6. Proof of Theorem \ref{['Thm4.4']}
  7. Appendix: The non-critical CKN-type inequalities

Key Result

Theorem 2.1

Assume that $1\le q<\infty$, $0<\eta\le +\infty$ and $w(t)\in V({\bf{R}}_+)$. Then, there exists a positive number $C=C(q, \eta, w)\ge 1$ such that for any $u\in C_c^1((0,\eta))$ we have Moreover either if $w(t)\in {W}_0({\bf{R}}_+)$ or if $w(t)\in {W}_\infty({\bf{R}}_+)$, $\eta=\infty$ and $\lim_{t\to\infty}w(t)=0$, then the best constant $C$ in (4.1Thm4.1) equals $1$.

Theorems & Definitions (23)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.1
  • Theorem 2.1
  • Remark 2.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: the non-degenerate condition
  • Theorem 2.2
  • Theorem 2.3
  • ...and 13 more