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 µ = (µ 1 , . . . , µ d ) ∈ M(R d , R d ) annihilating gradients of C (1) 0 (R d ) embedded in the natural way into C 0 (R d , R d ), i.e., such that i ∂ i f dµ i = 0 for f ∈ C (1) 0 (R d ), has Hausdorff dimension at least one. We provide examples which show both completeness and incompleteness of our results.

We introduce new spaces of martingales that behave like L_1 -based Sobolev spaces on \mathbb R^d . We prove a martingale analog of Van Schaftingen’s theorem and give sharp estimates on the lower Hausdorff dimension of the terminal distributions of these martingales. We also provide martingale analogs of trace theorems for Sobolev functions.

We prove that if pure derivatives with respect to all coordinates of a function on R n are signed measures, then their lower Hausdorff dimension is at least n − 1. The derivatives with respect to different coordinates may be of different order. * The authors are greatful to A.A. Logunov for suggestions concerning exposition of the material.

We investigate existence of a priori estimates for differential operators in L 1 norm: for anisotropic homogeneous differential operators T1, . . . , T ℓ , we study the conditions under which the inequalityholds true. We also discuss a similar problem for martingale transforms.