An estimate from below for the Markov constant of a Cantor repeller
Bellman approach to some problems in harmonic analysis
The stochastic optimal control uses the differential equation of Bellman and its solution - the Bellman function. Recently the Bellman function proved to be an efficient tool for solving some (sometimes old) problems in harmonic analysis.
Markov's property of the Cantor ternary set
We prove that the Cantor ternary set E satisfies the classical Markov inequality (see [Ma]): for each polynomial p of degree at most n (n = 0, 1, 2,...) (M) for x ∈ E, where M and m are positive constants depending only on E.
Rigidity of harmonic measure
Let J be the Julia set of a conformal dynamics f. Provided that f is polynomial-like we prove that the harmonic measure on J is mutually absolutely continuous with the measure of maximal entropy if and only if f is conformally equivalent to a polynomial. This is no longer true for generalized polynomial-like maps. But for such dynamics the coincidence of classes of these two measures turns out to be equivalent to the existence of a conformal change of variable which reduces the dynamical system...
Asymptotically holomorphic functions and translation invariant subspaces of weighted Hilbert spaces of sequences
Boundary Harnack principle for separated semihyperbolic repellers, harmonic measure applications.
Hilbert transform, Toeplitz operators and Hankel operators and invariant A weights.
In this paper, several sufficient conditions for boundedness of the Hilbert transform between two weighted L-spaces are obtained. Invariant A weights are obtained. Several characterizations of invariant A weights are given. We also obtain some sufficient conditions for products of two Toeplitz operators of Hankel operators to be bounded on the Hardy space of the unit circle using Orlicz spaces and Lorentz spaces.
Asymptotic values and the growth of analytic functions in spiral domains.
In this note we present a simple proof of a theorem of Hornblower which characterizes those functions analytic in the open unit disk having asymptotic values at a dense set in the boundary. Our method is based on a kind of ∂-mollification and may be of use in other problems as well.
Why the Riesz transforms are averages of the dyadic shifts?
The first author showed in [18] that the Hilbert transform lies in the closed convex hull of dyadic singular operators - so called dyadic shifts. We show here that the same is true in any Rn - the Riesz transforms can be obtained as the results of averaging of dyadic shifts. The goal of this paper is almost entirely methodological: we simplify the previous approach, rather than presenting the new one. [Proceedings of the 6th International Conference on...
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