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Vasily Vasyunin

Publications and source records attributed to Vasily Vasyunin.

At least 19 recordsLinked to original sources
arXiv2023

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.

math.CA↗
arXiv2023

Martingale transforms of bounded random variables and indicator functions of events

We provide sharp estimates for the distribution function of a martingale transform of the indicator function of an event. They are formulated in terms of Burkholder functions, which are reduced to the already known Bellman functions for extremal problems on $\mathrm{BMO}$. The reduction implicitly uses an unexpected phenomenon of automatic concavity for those Bellman functions: their concavity in some directions implies concavity with respect to other directions. A similar question for a martingale transform of a bounded random variable is also considered.

math.CA↗
arXiv2023

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.

math.CA↗
arXiv2022

On a Bellman function associated with the Chang--Wilson--Wolff theorem: a case study

In this paper we estimate the tail of distribution (i.e., the measure of the set $\{f\ge x\}$) for those functions $f$ whose dyadic square function is bounded by a given constant. In particular we get a bit better estimate than the estimate following from the Chang--Wilson--Wolf theorem. In the paper we investigate the Bellman function corresponding to the problem. A curious structure of this function is found: it has jumps of the first derivative at a dense subset of interval $[0,1]$ (where it is calculated exactly), but it is of $C^\infty$-class for $x>\sqrt3$ (where it is calculated up to a multiplicative constant). An unusual feature of the paper consists in the usage of computer calculations in the proof. Nevertheless, all the proofs are quite rigorous, since only the integer arithmetic was assigned to computer.

math.CA↗
arXiv2021

Sharp multiplicative inequalities with $\mathrm{BMO}$ $\mathrm{II}$

We find the best possible constant $C$ in the inequality $$\|φ\|_{L^r}^{\phantom{\frac{p}{r}}}\leq C\|φ\|_{L^p}^{\frac{p}{r}}\|φ\|_{\mathrm{BMO}}^{1-\frac{p}{r}}$$ for all possible values of parameters $p$ and $r$ such that $1 \le p < r < +\infty$. We employ the Bellman function technique to solve this problem. The Bellman function of three variables corresponding to this problem has a rather complicated structure, however, we managed to provide the explicit formulas for this function. First, we solve the problem on an interval and then transfer our results to the circle and the line. We also obtain explicit estimates in multi-dimensional cases.

math.CA↗
arXiv2021

Sharp moment estimates for martingales with uniformly bounded square functions

We provide sharp bounds for the exponential moments and $p$-moments, $1\leqslant p \leqslant 2$, of the terminate distribution of a martingale whose square function is uniformly bounded by one. We introduce a Bellman function for the corresponding extremal problem and reduce it to the already known Bellman function on $\mathrm{BMO}([0,1])$. In the case of tail estimates, a similar reduction does not work exactly, so we come up with a fine supersolution that leads to sharp tail estimates.

math.PR↗
arXiv2020

Sharp mutliplicative inequalities with $\mathrm{BMO}$ $\mathrm{I}$

We find the best possible constant $C$ in the inequality $\|φ\|_{L^r}\leq C\|φ\|_{L^p}^{\frac{p}{r}}\|φ\|_{\mathrm{BMO}}^{1-\frac{p}{r}}$, where $2 \leq r$ and $p < r$. We employ the Bellman function technique to solve this problem in the case of an interval and then transfer our results to the circle and the line.

math.CA↗
arXiv2019

The ${\rm BMO}\to{\rm BLO}$ action of the maximal operator on $α$-trees

We obtain the explicit upper Bellman function for the natural dyadic maximal operator acting from ${\rm BMO}(\mathbb{R}^n)$ into ${\rm BLO}(\mathbb{R}^n).$ As a consequence, we show that the ${\rm BMO}\to{\rm BLO}$ norm of the natural operator equals 1 for all $n,$ and so does the norm of the classical dyadic maximal operator. The main result is a partial corollary of a theorem for the so-called $α$-trees, which generalize dyadic lattices. The Bellman function in this setting exhibits an interesting quasi-periodic structure depending on $α,$ but also allows a majorant independent of $α,$ hence the dimension-free norm constant. We also describe the decay of the norm with respect to the difference between the average of a function on a cube and the infimum of its maximal function on that cube. An explicit norm-optimizing sequence is constructed.

math.CA↗
arXiv2016

On weak weighted estimates of martingale transform

We consider several weak type estimates for singular operators using the Bellman function approach. We disprove the $A_1$ conjecture of Muckenhoupt, which stayed open after Muckenhoupt--Wheeden's conjecture was disproved by Reguera--Thiele.

math.AP↗
arXiv2016

The John--Nirenberg constant of ${\rm BMO}^p$, $p>2$

This paper is a continuation of earlier work by the first author who determined the John--Nirenberg constant of ${\rm BMO}^p\big((0,1)\big)$ for the range $1\le p\le 2.$ Here, we compute that constant for $p>2.$ As before, the main results rely on Bellman functions for the $L^p$ norms of logarithms of $A_\infty$ weights, but for $p>2$ these functions turn out to have a significantly more complicated structure than for $1\le p\le 2.$

math.CA↗
arXiv2015

Cincinnati lectures on Bellman functions

In January-March 2011, the Department of Mathematical Science at the University of Cincinnati held a Taft Research Seminar "Bellman function method in harmonic analysis." The seminar was made possible by a generous grant from the Taft Foundation. The principal speaker at the seminar was Vasily Vasyunin. The local host and convener of the seminar was Leonid Slavin. The seminar was in effect a 10-week lecture- and discussion-based course. This manuscript represents a slightly revised content of those lectures. In particular, it includes some technical details that were omitted in class due to time constraints.

math.CA↗
arXiv2014

Inequalities for BMO on $α$-trees

We develop technical tools that enable the use of Bellman functions for BMO defined on $α$-trees, which are structures that generalize dyadic lattices. As applications, we prove the integral John--Nirenberg inequality and an inequality relating $L^1$- and $L^2$-oscillations for BMO on $α$-trees, with explicit constants. When the tree in question is the collection of all dyadic cubes in $\mathbb{R}^n,$ the inequalities proved are sharp. We also reformulate the John--Nirenberg inequality for the continuous BMO in terms of special martingales generated by BMO functions. The tools presented can be used for any function class that corresponds to a non-convex Bellman domain.

math.CA↗
arXiv2014

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 $Q$ is a cube in $\mathbb{R}^n$ and $h:[0,\infty)\to[0,\infty)$ is such that $h(t)\underset{t\to\infty}{\longrightarrow}\infty,$ then $$ \sup_{J \text{subcube} Q} \frac1{|J|}\int_J h(|φ-\frac1{|J|} \int_Jφ|)<\infty \Longrightarrow φ\in BMO(Q). $$ Under some additional assumptions on $h$ we obtain estimates on $\|φ\|_{BMO}$ in terms of the supremum above. We also show that even though the condition $h(t)\underset{t\to\infty}{\longrightarrow}\infty$ is not necessary for this implication to hold, it becomes necessary if one considers the dyadic BMO.

math.CA↗
arXiv2012

Bellman function for extremal problems in BMO

In this paper we develop the method of finding sharp estimates by using a Bellman function. In such a form the method appears in the proofs of the classical John--Nirenberg inequality and $L^p$ estimations of BMO functions. In the present paper we elaborate a method of solving the boundary value problem for the homogeneous Monge--Ampère equation in a parabolic strip for sufficiently smooth boundary conditions. In such a way we have obtained an algorithm of constructing an exact Bellman function for a large class of integral functionals in the BMO space.

math.AP↗
arXiv2011

Sharp L^p estimates on BMO

We construct the upper and lower Bellman functions for the $L^p$ (quasi)-norms of BMO functions. These appear as solutions to a series of Monge--Ampère boundary value problems on a non-convex plane domain. The knowledge of the Bellman functions leads to sharp constants in inequalities relating average oscillations of BMO functions and various BMO norms.

math.CA↗
arXiv2011

Burkholder's function via Monge--Ampère equation

We will show how to get Burkholder's function from \cite{Bu1} by using Monge-Ampére equation. This method is quite different from those in the series of Burkholder's papers \cite{Bu1}--\cite{Bu7}.

math.AP↗
arXiv2010

Some new Bellman functions and subordination by orthogonal martingales in $L^{p}, 1<p\le 2$

Given two martingales on the filtration generated by two dimensional Brownian motion, we want to estimate the $L^p$ norm of the subordinated one if we have some extra orthogonality property available. We construct several new Bellman functions, very different from Burkholder's function, and using them give an estimate of $L^p$ norm of a subordinated martingale, if the dominating martingale is orthogonal and $1<p\le 2$. We use Monge--Ampere equation to construct these Bellman functions.

math.PR↗