Singularity of vector valued measures in terms of Fourier transform
Journal article, 2006

We study how the singularity (in the sense of Hausdoiff 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(Rdbl; d , Rdbl; d ) annihilating gradients C 0 (1) (ℝ d ) embedded in the natural way into C 0 (ℝ d , ℝ d ), i.e., such that Σ ∫ ∂ i f d,μ i = 0 for f ∈ C 0 (1) (ℝ d ), has Hausdorff dimensional least one. We provide examples which show both completeness and incompleteness of our results. © 2006 Birkhäuser Boston. All rights reserved.

Author

Maria Roginskaya

University of Gothenburg

Chalmers, Mathematical Sciences

Michal Wojciechowski

Polish Academy of Sciences

Journal of Fourier Analysis and Applications

1069-5869 (ISSN)

Vol. 12 2 213-223

Subject Categories

Mathematics

DOI

10.1007/s00041-005-5030-9

More information

Latest update

10/30/2018