Pavel B. Zatitskiy

Geometry and dynamics of admissible metrics in measure spaces

We study a wide class of metrics in a Lebesgue space, namely the class of so-called admissible metrics. We consider the cone of admissible metrics, introduce a special norm in it, prove compactness criteria, define the ɛ-entropy of a measure space with an admissible metric, etc. These notions and related results are applied to the theory of transformations with invariant measure; namely, we study the asymptotic properties of orbits in the cone of admissible metrics with respect to a given transformation or a group of transformations. The main result of this paper is a new discreteness criterion for the spectrum of an ergodic transformation: we prove that the spectrum is discrete if and only if the ɛ-entropy of the averages of some (and hence any) admissible metric over its trajectory is uniformly bounded.

Weak integral conditions for BMO

We study the question of how much one can weaken the defining condition of BMO. Specifically, we show that if

The Tamarkin equiconvergence theorem and a first-order trace formula for regular differential operators revisited

Q

Formula of regularized traces

is a cube in

ON EMBEDDINGS OF FINITE METRIC SPACES IN l n

\mathbb{R}^n

Bellman function for extremal problems in BMO

and

Bellman VS. Beurling: sharp estimates of uniform convexity for ^{} spaces

h:[0,\infty)\to[0,\infty)

Sharp estimates of integral functionals on classes of functions with small mean oscillation

is such that

Virtual continuity of measurable functions of several variables and embedding theorems

h(t)\underset{t\to\infty}{\longrightarrow}\infty,

Theory of locally concave functions and its applications to sharp estimates of integral functionals

then