[Paper Review] Weighted Sobolev Inequalities via the Meyers--Ziemer Framework: Measures, Isoperimetric Inequalities, and Endpoint Estimates

Simon Bortz, OFS+3 more|arXiv (Cornell University)|Mar 5, 2026

Nonlinear Partial Differential Equations0 citations

TL;DR

The paper develops a generalized endpoint Sobolev inequality for measures using a Meyers–Ziemer–type framework with a maximal function, and derives consequences for BV, capacity, isoperimetry, and endpoint estimates for fractional operators, including two-weight Sobolev inequalities.

ABSTRACT

We establish a new global endpoint Sobolev inequality for measures that extends the classical theorem of Meyers-Ziemer by placing a maximal function on the right-hand side. This result has several significant consequences. It extends naturally to functions of weighted bounded variation and yields corresponding capacity and isoperimetric inequalities. The inequality is also closely connected to endpoint estimates for fractional operators, including bounds for fractional maximal functions and Hardy space endpoint estimates for the Riesz potential. Our main inequality yields a family of endpoint inequalities, characterized in terms of subrepresentation formulas, Lorentz space improvements, and isoperimetric inequalities for measures and bounded open sets. When one moves away from the endpoint to $p>1$, the analogous inequalities no longer hold in general; however, we identify a sharp bumped maximal function for which the corresponding non-endpoint inequality is valid. Finally, we show that this framework yields new $(p,p)$ two-weight Sobolev inequalities.

Motivation & Objective

• Generalize the Meyers–Ziemer endpoint Sobolev inequality to measures with a maximal function weight.
• Derive BV, capacity, and isoperimetric inequalities in the weighted setting.
• Connect endpoint Sobolev inequalities to endpoint estimates for fractional operators and Riesz potentials.
• Establish two-weight Sobolev inequalities and discuss sharpness of bump conditions.

Proposed method

• Prove a global endpoint Sobolev inequality for measures featuring a right-hand side maximal function: (23) ∫ |u| dμ ≤ C ∫ |∇u| M1μ dx for Lip_c functions.
• Show equivalence between finiteness of Mαμ and Hausdorff measure via Theorem 2.2, and that M1μ is an A1 weight when finite a.e.
• Derive weighted BV, capacity, and isoperimetric inequalities from (23), including Corollary 2.3 and Corollary 2.7.
• Establish endpoint estimates for Riesz potentials and transforms, including Theorem 2.8 and Theorem 2.11, relating I1 and R via M1μ.
• Discuss Lorentz space refinements and extensions to Lorentz-scale Sobolev inequalities (Corollary 2.12).
• Present two-weight Sobolev inequalities and bump conditions, including optimal results in the diagonal case (p=p).

Experimental results

Research questions

• RQ1 What is the appropriate endpoint Sobolev inequality for measures that naturally extends Meyers–Ziemer?
• RQ2 How does introducing M1μ as a weight on the gradient side affect BV, capacity, and isoperimetric inequalities?
• RQ3 What are the endpoint bounds for fractional operators (Iα, Mα) in this framework, and how do Riesz transforms feature in these estimates?
• RQ4 Can the framework yield two-weight (p,p) Sobolev inequalities with sharp bump conditions?
• RQ5 How do Lorentz refinements arise in this endpoint framework and what are their implications for weighted Sobolev inequalities?

Key findings

• A global endpoint Sobolev inequality for measures holds with a right-hand side maximal function: ∫ |u| dμ ≤ C ∫ |∇u| M1μ dx.
• If Mαμ is finite somewhere, then it is finite a.e. outside an Hn−α-null set and Mαμ ∈ A1; this underpins weight behavior in the framework.
• The framework yields BV(w) extensions, w-perimeter representations, and weighted isoperimetric inequalities (Corollaries 2.3–2.7).
• Endpoint bounds for I1 and the vector R f are established: ∥I1 f∥L1(μ) ≤ C ∥R f∥L1(M1 μ) and related Hardy space bounds (Theorem 2.8, Theorem 2.11).
• Lorentz-space endpoint refinements are obtained, e.g., ∥u∥L n/(n−1),1(μ) ≤ C ∫ |∇u| (Mα μ)1/q dx (Corollary 2.12).
• Two-weight Sobolev inequalities (p,p) with bump conditions are analyzed, including optimal results in the diagonal case (p=q) under refined bump assumptions.