From generalized Poincaré to Poincaré-Sobolev inequalities via self-improving methods

Abstract: We establish several improvements to the main results of [PR19] and [CP21], refining the seminal self-improving method for generalized Poincaré inequalities from [FPW98, MP98]. These results, together with various related applications, stem from a general self-improving property for functions satisfying the local inequality $\frac{1}{|Q|}\int_Q |f(x)-f_Q|\,dx \le a(Q)$ for all cubes $Q\subset\mathbb{R}<sup>n$. The functional $a$ is assumed to obey a specific discrete geometric summability condition. By restricting our focus to axis-parallel cubes in $\mathbb{R}<sup>n$, this geometric setting allows us to obtain sharper estimates than those available in more general metric measure spaces.

• The paper establishes sharp self-improving phenomena linking generalized Poincaré inequalities with weighted Sobolev estimates via discrete geometric functionals.
• It proves that local oscillation inequalities on axis-parallel cubes self-improve to Lorentz weak-type or exponential integrability under specific weight and summability conditions.
• The work extends the methods to higher-order derivatives, vector-valued mappings, and rectangular domains, impacting regularity theories for degenerate elliptic PDEs.

Self-Improvement from Generalized Poincaré to Poincaré-Sobolev Inequalities

Introduction and Context

This paper presents a refined analysis of self-improving phenomena in generalized Poincaré inequalities, linking them to sharp weighted Poincaré-Sobolev estimates using discrete geometric criteria for oscillation functionals. Given the modern role of such inequalities in harmonic analysis, geometric measure theory, and regularity theory for PDEs, the authors' focus on axis-parallel cubes in $\mathbb{R}^n$ enables advances beyond the general setting of spaces of homogeneous type. The improved self-improving mechanisms here are fundamentally geometric, depending on discrete summability conditions on certain functionals and weights, generalizing the classical $A_p$ theory and allowing sharp quantification into Lorentz and Orlicz-scale target norms.

Self-Improving Mechanisms and Main Theorems

A central aspect is the consideration of local oscillation inequalities,

$\fint_Q |f(x) - f_Q|\,dx \leq a(Q)$

for all cubes $Q$, with $a$ a positive functional satisfying geometric summability such as the $D_p(w)$ or the strictly stronger $SD_p^s(w)$ condition, involving weighted measures and subcube decomposition. These conditions govern whether the oscillation improves from $L^1$ to a scale of higher integrability norms under minimal assumptions, generalizing the classical John-Nirenberg and Poincaré approaches.

Main theorems:

Given a functional $a \in SD_p^s(w)$, and $w\in A_r$, the authors prove:

• If $A_p$0, the oscillation self-improves to a weighted Lorentz weak-$A_p$1 inequality,

$A_p$2

with sharp exponent $A_p$3 defined by $A_p$4.

• If $A_p$5, only exponential integrability (Luxemburg norm for Orlicz spaces) can be deduced,

$A_p$6

These mechanisms produce sharp dependence on $A_p$7 and $A_p$8, with constants further controlled by the discrete geometric parameters of $A_p$9 and the dimension $\fint_Q |f(x) - f_Q|\,dx \leq a(Q)$0 (2606.08556).

Weighted Poincaré–Sobolev and Fractional Inequalities

The authors derive optimal weighted Poincaré–Sobolev inequalities as corollaries, recovering and sharpening the previous best results for the range of exponents and weight classes. Notably:

• For $\fint_Q |f(x) - f_Q|\,dx \leq a(Q)$1, $\fint_Q |f(x) - f_Q|\,dx \leq a(Q)$2, and the local cube,

$\fint_Q |f(x) - f_Q|\,dx \leq a(Q)$3

where $\fint_Q |f(x) - f_Q|\,dx \leq a(Q)$4.

• For fractional Sobolev settings:

$\fint_Q |f(x) - f_Q|\,dx \leq a(Q)$5

where $\fint_Q |f(x) - f_Q|\,dx \leq a(Q)$6 reflects the scaling $\fint_Q |f(x) - f_Q|\,dx \leq a(Q)$7.

The log-convexity and Lorentz space improvement is realized via truncation and good-lambda methods, extending classical embeddings and endpoint Trudinger/Moser-type results to the weighted, discrete-summability-driven regime.

Novelty: Counterexamples and Sharpness

An explicit construction of counterexamples is provided to demonstrate the impossibility of further improvement for this self-improving phenomenon outside the stated exponent ranges, even for smooth weights just outside the $\fint_Q |f(x) - f_Q|\,dx \leq a(Q)$8 regime or for exponents $\fint_Q |f(x) - f_Q|\,dx \leq a(Q)$9. This establishes the sharpness of the main theorems and clarifies where the transition from power-scale (Sobolev) to exponential-scale (Orlicz/Trudinger) integrability occurs. Additionally, a necessity result shows that the $Q$0 property for weights is required for the $Q$1-based self-improving paradigm, decisively refuting any possibility of extension beyond Muckenhoupt-type weights (2606.08556).

Extensions: Higher-Order, Vector-Valued, and Rectangular Geometries

The argument structure encompasses extensions to:

• Higher-order derivatives by replacing averages with degree-$Q$2 polynomial projections, producing inequalities optimal for the Sobolev space $Q$3 and its weighted and Lorentz targets.
• Vector-valued mappings taking values in $Q$4, showing the self-improvement persists when oscillations and gradients are measured with respect to the Banach norm.
• Rectangular domains (axis-aligned products), relevant for multi-parameter harmonic analysis.

These generalizations confirm that the underlying geometric-dominated machinery remains robust in these broader functional and structural settings.

Implications and Outlook

The technical development in this work clarifies both the mechanistic (geometric-discrete versus analytic-representation driven) and the pragmatic (sharpness in weights, exponents, domain geometry, and function class) aspects of self-improving inequalities for local oscillations. This mechanism yields optimal estimates underpinning regularity theories for degenerate and weighted elliptic PDEs, including those with $Q$5 or $Q$6-ellipticity, and supports subsequent analysis for non-homogeneous or product-space settings.

Further directions initiated include:

• Tight integration with singular integral and sparse domination theory, as sparse bounds underlie the sharp dependence on weight constants.
• Improved endpoint and vector-valued regularity results for divergence-form equations and systems, including in degenerate and anisotropic settings.
• Analysis of self-improvability for oscillations with respect to more general metrics or measures, particularly beyond the classical Euclidean/doubling context.

Conclusion

This paper establishes comprehensive, sharp self-improving inequalities that bridge local oscillation control and higher integrability/regularity for weighted function spaces on cubes, with geometric, Lorentz, and Orlicz refinements. The general mechanism clarifies both necessary and sufficient conditions in terms of discrete geometric functionals and weight classes, and robustly extends to higher-order, vector-valued, paired-weight, and rectangular contexts. These advances provide foundational tools for the analysis of degenerate elliptic PDEs, the theory of weights, and further developments in geometric and harmonic analysis (2606.08556).