Maria Roginskaya

A Fourier series formula for energy of measures with applications to Riesz products

In this paper we derive a formula relating the energy and the Fourier transform of a finite measure on the d-dimensional torus which is similar to the well-known formula for measures on R d . We apply the formula to obtain estimates on the Hausdorff dimension of Riesz product measures. These give improvements on the earlier, classical results which were based on completely different techniques.

A general approach to Read's type constructions of operators without non-trivial invariant closed subspaces

We present a general method for constructing operators without non-trivial invariant closed subsets on a large class of non-reflexive Banach spaces. In particular, our approach unifies and generalizes several constructions due to Read of operators without non-trivial invariant subspaces on the spaces l(1), c(0) or circle plus(l2) J, and without non-trivial invariant subsets on l(1). We also investigate how far our methods can be extended to the Hilbertian setting, and construct an operator on a quasi-reflexive dual Banach space which has no non-trivial w*-closed invariant subspace.

The energy of signed measures

We generalize the concept of energy to complex measures of finite variation. We show that although the energy dimension of a measure can exceed that of its total variation, it is always less than the Hausdorff dimension of the measure. As an application we prove a variant of the uncertainty principle.

On L p -improving measures

We give criteria for establishing that a measure is Lp-improving. Many Riesz product measures and Cantor measures satisfy this criteria, as well as certain Markov measures

Some Supports of Fourier Transforms of Singular Measures are not Rajchman

The notion of Riesz sets tells us that a support of Fourier transform of a measure with non-trivial singular part has to be large. The notion of Rajchman sets tells us that if the Fourier transform tends to zero at infinity outside a small set, then it tends to zero even on the small set. Here we present a new angle of an old question: Whether every Rajchman set should be Riesz.

Conjugacy of real diffeomorphisms. A survey

Given a group G, the conjugacy problem in G is the problem of giving an effective procedure for determining whether or not two given elements f, g of G are conjugate, i.e. whether there exists h belonging to G with fh = hg. This paper is about the conjugacy problem in the group Diffeo(I) of all diffeomorphisms of an interval I in R. There is much classical work on the subject, solving the conjugacy problem for special classes of maps. Unfortunately, it is also true that many results and arguments known to the experts are difficult to find in the literature, or simply absent. We try to repair these lacunae, by giving a systematic review, and we also include new results about the conjugacy classification in the general case.

Asymptotic Properties of Harmonic and M-Harmonic Functions near the Boundary of the Unit Ball

It is shown that decompositions of a plurisubharmonic measure on a sphere of ℂn with respect to the Hausdorff dimension scales related to the Euclidean and Korányi metrics coincide under a linear correspondence of indices. Bibliography: 7 titles.