Some extremal problems for martingale transforms, I

With this paper, we begin a series of studies of extremal problems for estimating distributions of martingale transforms of bounded martingales. The Bellman functions corresponding to such problems are pointwise minimal diagonally concave functions on a horizontal strip, satisfying certain given boundary conditions. We describe the basic structures that arise when constructing such functions and present a solution in the case of asymmetric boundary conditions and a sufficiently small width of the strip

A two-line representation of stationary measure for open TASEP

We show that the stationary measure for the totally asymmetric simple exclusion process on a segment with open boundaries is given by a marginal of a two-line measure with a simple and explicit description. We use this representation to analyze asymptotic fluctuations of the height function near the triple point for a larger set of parameters than was previously studied. As a second application, we determine a single expression for the rate function in the large deviation principle for the height function in the fan and in the shock region. We then discuss how this expression relates to the expressions available in the literature

Monotone rearrangement in averaging classes

We consider a general collection of function classes on the interval $[0,1]$ defined in terms of certain averages and show that monotone rearrangement does not increase the class constant in each case. The formulation includes BMO and $A_2$ with a special choice of the norm and, respectively, characteristic.Comment: 8 page

On the relations between Auerbach or almost Auberbach Markushevich systems and Schauder bases

We establish that the summability of the series $\sum\varepsilon_n$ is the necessary and sufficient criterion ensuring that every $(1+\varepsilon_n)$ Markushevich basis in a separable Hilbert space is a Riesz basis. Further we show that if $n\varepsilon_n\to \infty$, then in $\ell_2$ there exists a $(1+\varepsilon_n)$ Markushevich basis that under any permutation is non-equivalent to a Schauder basis. We extend this result to any separable Banach space. Finally we provide examples of Auerbach bases in 1-symmetric separable Banach spaces whose no permutations are equivalent to any Schauder basis or (depending on the space) any unconditional Schauder basis.Comment: We improved the presentation of the proof of Theorem 1.3 and corrected some typos. There are no essential changes except the removal of the last sentence of Example 8.5 which was not correct. The main statement of Example 8.5 remains unchanged. We slightly reformulated the Acknowledgment

New Bellman induction and a weak version of BMO\mathrm{BMO}

We enlarge the area of applicability of the Bellman function method to estimates in the spirit of the John--Nirenberg inequality abandoning certain convexity assumptions. As an application, we consider a characteristic of a function that is much smaller than the $\mathrm{BMO}$ norm, but whose finiteness leads to the exponential integrability of the function.Comment: 19 page

Bellman functions on simple non-convex domains in the plane

The present paper provides a generalization of the previous authors' work on Bellman functions for integral functionals on $\mathrm{BMO}$. Those Bellman functions are the minimal locally concave functions on parabolic strips in the plane. Now we describe the algorithm for constructing minimal locally concave functions on a planar domain that is a difference of two unbounded convex domains. This leads to many sharp estimates for functions in the classes like $\mathrm{BMO}$, $A_p$, or the Gehring classes.Comment: 106 pages. arXiv admin note: text overlap with arXiv:1510.0101