On dimension and regularity of vector-valued measures under Fourier analytic constraints

Abstract

In this paper, we investigate fine properties of measures belonging to a certain class of vector-valued measures arising as a natural generalization of gradients of BV functions. Our results concern dimensional estimates and rectifiability. We propose two conditions on the Fourier transform under which such measures cannot be very singular. The first one is related to the Fourier antisymmetry. The second one is a modification of the 2-wave cone condition for $\mathcal{A}$-free measures.