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Degenerate Poincaré-Sobolev inequalities via fractional integration

Alejandro Claros

TL;DR

This paper advances the theory of weighted Poincaré-Sobolev inequalities in degenerate elliptic settings by establishing sharp, locally conservative bounds for the Riesz potential with $A_p$ weights. The authors develop a Hedberg-type framework to obtain weighted local bounds for $I_\alpha$, introduce mixed $A_p$–$A_\infty$ constants, and extend results to $p=1$, high-order derivatives, and fractional settings with BBM-type gain factors. A key contribution is resolving a Pérez–Rela conjecture for the $A_1$ case and defining an optimal weighted Sobolev exponent $p^*_w$ that governs the range of valid exponents, with explicit counterexamples illustrating sharpness. The results yield improved constants and exponent ranges for weighted Poincaré-Sobolev inequalities, thereby strengthening the applicability of these inequalities to regularity theory for degenerate PDEs and to related fractional Sobolev frameworks.

Abstract

We present a local weighted estimate for the Riesz potential in $\mathbb{R}^n$, which improves the main theorem of Alberico, Cianchi, and Sbordone [C. R. Math. Acad. Sci. Paris \textbf{347} (2009)] in several ways. As a consequence, we derive weighted Poincaré-Sobolev inequalities with sharp dependence on the constants. We answer positively to a conjecture proposed by Pérez and Rela [Trans. Amer. Math. Soc. 372 (2019)] related to the sharp exponent in the $A_1$ constant in the $(p^*,p)$ Poincaré-Sobolev inequality with $A_1$ weights. Our approach is versatile enough to prove Poincaré-Sobolev inequalities for high-order derivatives and fractional Poincaré-Sobolev inequalities with the BBM extra gain factor $(1-δ)^{1/p}$. In particular, we improve one of the main results from Hurri-Syrjänen, Martínez-Perales, Pérez, and Vähäkangas [Int. Math. Res. Not. 20 (2023)].

Degenerate Poincaré-Sobolev inequalities via fractional integration

TL;DR

This paper advances the theory of weighted Poincaré-Sobolev inequalities in degenerate elliptic settings by establishing sharp, locally conservative bounds for the Riesz potential with weights. The authors develop a Hedberg-type framework to obtain weighted local bounds for , introduce mixed – constants, and extend results to , high-order derivatives, and fractional settings with BBM-type gain factors. A key contribution is resolving a Pérez–Rela conjecture for the case and defining an optimal weighted Sobolev exponent that governs the range of valid exponents, with explicit counterexamples illustrating sharpness. The results yield improved constants and exponent ranges for weighted Poincaré-Sobolev inequalities, thereby strengthening the applicability of these inequalities to regularity theory for degenerate PDEs and to related fractional Sobolev frameworks.

Abstract

We present a local weighted estimate for the Riesz potential in , which improves the main theorem of Alberico, Cianchi, and Sbordone [C. R. Math. Acad. Sci. Paris \textbf{347} (2009)] in several ways. As a consequence, we derive weighted Poincaré-Sobolev inequalities with sharp dependence on the constants. We answer positively to a conjecture proposed by Pérez and Rela [Trans. Amer. Math. Soc. 372 (2019)] related to the sharp exponent in the constant in the Poincaré-Sobolev inequality with weights. Our approach is versatile enough to prove Poincaré-Sobolev inequalities for high-order derivatives and fractional Poincaré-Sobolev inequalities with the BBM extra gain factor . In particular, we improve one of the main results from Hurri-Syrjänen, Martínez-Perales, Pérez, and Vähäkangas [Int. Math. Res. Not. 20 (2023)].

Paper Structure

This paper contains 9 sections, 21 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Statement of the results
  3. Weighted local bounds for fractional integrals with $A_p$ weights
  4. Applications to Poincaré-Sobolev inequalities
  5. High order derivatives
  6. Optimality of the weighted Sobolev exponent
  7. Preliminaries and known results
  8. Weighted Hedberg's inequality and consequences
  9. Poincaré-Sobolev inequalities and weighted Sobolev exponent

Key Result

Theorem 1.1

Let $n\ge 2$, $\alpha \in (0,n)$ and $1<p<\frac{n}{\alpha}$. Let $w\in A_p$ and given $r\ge 1$ consider $q$ defined by the relation Then, there exist positive constants $k=k(p,n)$ and $C=C(\alpha, p, n)$ such that if then for any ball $B\subset \mathbb{R}^n$ of radius $r(B)$ and every function $f\in L^p(B, w)$ (continued by $0$ outside $B$). Moreover, the exponent $\frac{nr-\alpha }{nr(p-1)}$ o

Theorems & Definitions (40)

  • Theorem 1.1: ACS
  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5
  • Remark 2.6
  • Theorem 2.7
  • Remark 2.8
  • Theorem 2.9
  • ...and 30 more