A look into some of the fine properties of functions with bounded 𝒜-variation

¹ Dipartimento di Matematica, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy
² Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland

^* Corresponding author: adolfo.rabasa@unipi.it

Received: 3 May 2023
Accepted: 2 January 2025

Abstract

We establish certain fine properties for functions of bounded 𝒜-variation known in the classical BV setting. Here, 𝒜 is a kth order constant-coefficient homogeneous linear differential operator with a finite-dimensional kernel (also known as a complex-elliptic operator). We prove that if 𝒜u can be represented by a finite Radon measure, then the potential u has one-sided L^p-approximate limits on Lipschitz hypersurfaces, and, more generally, on countably rectifiable sets of codimension one. We use this to give pointwise characterizations of the (functional) interior and exterior traces. We also establish a quantitative scale-dependent continuity result, which allows us to prove that the Lebesgue discontinuity set has zero (n − 1)-dimensional Riesz capacity. Lastly, we introduce a decomposition that reduces the complexity of analyzing kth-order operators to that of first-order methods and allows us to establish the kth order L^p-differentiability of BV^𝒜 maps.