Hölder regularity of the solutions of Fredholm integral equations on upper Ahlfors regular sets

TL;DR

The paper addresses Hölder regularity of solutions to Fredholm integral equations of the second kind on upper Ahlfors regular sets, including nondoubling measures, by developing a metric-measure framework. It introduces and analyzes potential-type kernel classes ${\mathcal{K}}_{s,X\times Y}$ and their composites on upper Ahlfors regular sets, proves continuity of composite kernels and extensions to $Y^2$, and then establishes regularity results for solutions: from mere continuity of data $g$ to generalized Hölder continuity of the solution $\mu$ (with explicit modulus $\varpi$). It also specializes to boundary-integral settings on Lipschitz and $C^1$ domains, deriving concrete Hölder or logarithmic-type regularity for $\mu$ when the boundary is smooth enough. Collectively, the results extend classical potential-theoretic regularity theory from doubling/Lebesgue settings to nondoubling, upper-regular metric-measure spaces, with implications for boundary value problems and elliptic PDEs in irregular domains.

Abstract

We extend to the context of metric measured spaces, with a measure that satisfies upper Ahlfors growth conditions the validity of (generalized) Hölder continuity results for the solution of a Fredholm integral equation of the second kind. Here we note that upper Ahlfors growth conditions include also cases of nondoubling measures.

Theorems & Definitions (18)

• Remark 2.2
• Definition 3.2
• Definition 3.3
• Remark 3.4
• Lemma 3.5
• Lemma 3.6
• Lemma 3.7
• Lemma 3.9
• Proposition 4.1
• Theorem 4.9