Singularity of Vector Valued Measures in Terms of Fourier Transform

Abstract

We study how the singularity (in the sense of Hausdorff dimension) of a vector valued measure can be affected by certain restrictions imposed on its Fourier transform. The restrictions, we are interested in, concern the direction of the (vector) values of the Fourier transform. The results obtained could be considered as a generalizations of F. and M. Riesz theorem, however a phenomenon, which have no analogy in the scalar case, arise in the vector valued case. As an example of application, we show that every measure from \(\mu=(\mu_1,\dots,\mu_d)\in M({\mathbb R}^d,{\mathbb R}^{d})\) annihilating gradients of \(C^{(1)}_0({\mathbb R}^d)\) embedded in the natural way into \(C_0({\mathbb R}^d,{\mathbb R}^{d}),\) i.e., such that \(\sum_i\int\partial^if\,d\mu_i=0\,\) for \(f\in C^{(1)}_0({\mathbb R}^d)\), has Hausdorff dimension at least one. We provide examples which show both completeness and incompleteness of our results.