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 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.

On finite configurations in the spectra of singular measures

“…One of the results proved in [RW06] (Theorem 3 therein) is a theorem which delivers a criterion when the dimension of such vector measure is at least one.…”

“…In the paper [RW06] the problem of dimensional estimates for vector measures was considered under restrictions on the phase of the Fourier transform. More precisely, let φ : S n´1 Ñ G C pd, Eq be a continuous function, called henceforth a bundle 2 , taking values in the Grassmannian of d-dimensional (complex) subspaces of some finite dimensional complex vector space E. The class of measures M φ pR n , Eq, subordinated to φ, is defined as follows.…”

Martingale approach to Sobolev embedding theorems

“…We believe that this result may be generalized to the setting of BV-spaces defined above. We cite a result from [21], where condition (1.2) plays the central role.…”

Dimension of gradient measures

“…However, can one say something if this condition does not hold? We cite a simpler version Theorem 3 from [11]. Theorem 1.…”

“…In this particular case, Theorem 1 is weaker (we get only dimension 1). One can make a courageous conjecture (Conjecture 1 in [11]). Conjecture 1.…”