Slicing and fine properties for functions with bounded $\mathcal A$-variation

Abstract

We study the slicing and fine properties of functions in BV A , the space of functions of bounded A-variation. Here, A is a homogeneous linear differential operator with constant coefficients (of arbitrary order). Our main result is the characterization of all A satifying the following one-dimensional structure theorem: every u ∈ BV A can be sliced into one-dimensional BV-sections. Moreover, decomposing Au into an absolutely continuous part A a u, a Cantor part A c u and a jump part A j u, each of these measures can be recovered from the corresponding classical D a , D c and D j BV -derivatives of its one-dimensional sections. By means of this result, we are able to analyze the set of Lebesgue points as well as the set of jump points where these functions have approximate one-sided limits. Thus, proving a structure and fine properties theorem in BV A . Our results extend most of the classical fine properties of BV (and all of those known for BD) to BV A . Applications of our results are discussed for operators that are not covered by the existing theory. Contents 1. Introduction 2.1. Slicing of first-order operators 2.2. Rectifiability of A-gradient measures 2.3. Structural and fine properties for first-order elliptic operators 2.4. Statements for higher-order operators 3. Preliminaries 3.1. Disintegration into one-dimensional sections 3.2. The jump set and approximate continuity 3.3. Symbolic calculus 3.4. Traces of complex elliptic operators 4. Slicing theory for first-order operators 4.1. The rank-one property 4.2. Proof of the sectional representation theorem 4.3. Algebraic constructions 4.4. Stability properties of co-dimension one slicing 4.5. Slices of arbitrary co-dimension 4.6. Polarization properties 5. Analysis of Lebesgue points 6. Proof of the one-dimensional structure theorem

Frostman lemma revisited

“…Let ϕ be a radially symmetric, radially non-increasing function supported in the unit ball. Assume also ϕ(x) = 1 when |x| 3 4 . Let ψ be equivalent to ϕ.…”

“…Theorem 2.2. Let ϕ be a bounded function supported in the unit ball and such that ϕ(x) 1 when |x| 3 4 . Let µ be an R-valued signed Radon measure and let g be a regular function.…”

“…) is a radially decreasing non-negative function supported in the unit ball. Assume that ϕ decreases as the radius grows and ϕ(x) = 1 when |x| ≤ 3 4 . Let µ be a measure such that for every collection B r j (x j ) of d-dimensional balls such that B 3r j (x j ) are disjoint, the estimate…”

“…In recent years, there is an increasing interest in the geometry of measures satisfying PDE or Fourier constraints, like gradients of functions of bounded variation, or divergence free measures, or, for example, the generalized gradient measures as above. We refer the reader to the papers [2,3,5,4,8,13], to mention a few (see [17] for limiting Sobolev inequalities for vectorial differential operators that are closely related to the topic). Of primary interest is the dimension problem: what is the lowest possible Hausdorff dimension of a measure solving a specific PDE?…”

Dimensional estimates and rectifiability for measures satisfying linear PDE constraints

“…A parade of examples of constant-coefficient elliptic operators can be found in [30] (see also [6] where also other less known examples are discussed) Elliptic operators possess an annihilator operator A, that is, a constant-coefficient linear PDE operator from V to V such that…”