arXiv subjects

Vasily Vasyunin

Publications and source records attributed to Vasily Vasyunin.

21 records · Page 2Linked to original sources

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↗

arXiv2010

Invertibility threshold for $H^\infty$ trace algebras, and effective matrix inversions

For a given $δ$, $0<δ<1$, a Blaschke sequence $σ=\{λ_j\}$ is constructed such that every function $f$, $f\in H^\infty$, having $δ<δ_f=\inf_{λ\inσ}|f(λ)|\le\|f\|_\infty\le1$ is invertible in the trace algebra $H^\infty|σ$ (with a norm estimate of the inverse depending on $δ_f$ only), but there exists $f$ with $δ=δ_f\le\|f\|_\infty\le1$, which does not. As an application, a counterexample to a stronger form of the Bourgain--Tzafriri restricted invertibility conjecture for bounded operators is exhibited, where an ``orthogonal (or unconditional) basis'' is replaced by a ``summation block orthogonal basis''.

math.FA↗

arXiv2008

Monge--Ampère equation and Bellman optimization of Carleson Embedding Theorems

Monge--Ampère equation plays an important part in Analysis. For example, it is instrumental in mass transport problems. On the other hand, the Bellman function technique appeared recently as a way to consider certain Harmonic Analysis problems as the problems of Stochastic Optimal Control. This brings us to Bellman PDE, which in stochastic setting is often a Monge--Ampère equation or its close relative. We explore the way of solving Monge--Ampère equation by a sort of method of characteristics to find the Bellman function of certain classical Harmonic Analysis problems, and, therefore, of finding full structure of sharp constants and extremal sequences for those problems.

math.AP↗