📝 SummarXiv

Abstract

We investigate the limiting behavior of Besov seminorms and nonlocal perimeters in Dunkl theory. The present work generalizes two fundamental results: the Maz'ya--Shaposhnikova formula for Gagliardo seminorms and the asymptotics of (relative) fractional $s$-perimeters. Our main contributions are twofold. First, we establish a dimension-free Maz'ya--Shaposhnikova formula via a novel, robust approach that avoids reliance on the density property of Besov spaces, offering broader applicability. Second, we prove limit formulas for nonlocal perimeters relative to bounded open sets $\Omega$, removing boundary regularity assumptions in the forward direction, while introducing a weakened regularity condition on $\partial\Omega$ (admitting fractal boundaries) for the converse, a significant improvement over existing requirements. To the best of our knowledge, the results in this second part are new even in the classic Laplacian setting.