arXiv subjects

Adam Osȩkowski

Publications and source records attributed to Adam Osȩkowski.

4 recordsLinked to original sources

arXiv2019

The ${\rm BMO}\to{\rm BLO}$ action of the maximal operator on $\alpha$-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 $\alpha$-trees, which generalize dyadic lattices. The Bellman function in this setting exhibits an interesting quasi-periodic structure depending on $\alpha,$ but also allows a majorant independent of $\alpha,$ 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↗

arXiv2012

Sharp maximal inequalities for the moments of martingales and non-negative submartingales

In the paper we study sharp maximal inequalities for martingales and non-negative submartingales: if $f$, $g$ are martingales satisfying \[|\mathrm{d}g_n|\leq|\mathrm{d}f_n|,\qquad n=0,1,2,...,\] almost surely, then \[\Bigl\|\sup_{n\geq0}|g_n|\Bigr\|_p\leq p\|f\|_p,\qquad p\geq2,\] and the inequality is sharp. Furthermore, if $\alpha\in[0,1]$, $f$ is a non-negative submartingale and $g$ satisfies \[|\mathrm{d}g_n|\leq|\mathrm{d}f_n|\quad and\quad |\mathbb{E}(\mathrm{d}g_{n+1}|\mathcal {F}_n)|\leq\alpha\mathbb{E}(\mathrm{d}f_{n+1}|\mathcal{F}_n),\qquad n=0,1,2,...,\] almost surely, then \[\Bigl\|\sup_{n\geq0}|g_n|\Bigr\|_p\leq(\alpha+1)p\|f\|_p,\qquad p\geq2,\] and the inequality is sharp. As an application, we establish related estimates for stochastic integrals and It\^{o} processes. The inequalities strengthen the earlier classical results of Burkholder and Choi.

math.ST↗

arXiv2011

Noncommutative maximal inequalities associated with convex functions

We prove several noncommutative maximal inequalities associated with convex functions, including a Doob type inequality for a convex function of maximal operators on noncommutative martingales, noncommutative Dunford-Schwartz and Stein maximal ergodic inequalities for a convex function of positive and symmetric positive contractions. The key ingredient in our proofs is a Marcinkiewicz type interpolation theorem for a convex function of maximal operators in the noncommutative setting, which we establish in this paper. These generalize the results of Junge and Xu in the $L^p$ case to the case of convex functions.

math.OA↗

arXiv2009

Sharp weak-type inequalities for differentially subordinated martingales

Let $M,N$ be real-valued martingales such that $N$ is differentially subordinate to $M$. The paper contains the proofs of the following weak-type inequalities: (i) If $M\geq0$ and $0<p\leq1$, then \[\Vert N\Vert_{p,\infty}\leq2\Vert M\Vert_p\] and the constant is the best possible. (ii) If $M\geq0$ and $p\geq2$, then \[\Vert N\Vert_{p,\infty}\leq\frac{p}{2}(p-1)^{-1/p}\Vert M\Vert_p\] and the constant is the best possible. (iii) If $1\leq p\leq2$ and $M$ and $N$ are orthogonal, then \[\Vert N\Vert_{p,\infty}\leq K_p\Vert M\Vert_p,\] where \[K_p^p=\frac{1}{Γ(p+1)}\cdot\biggl(\fracπ{2}\biggr)^{p-1}\cdot\frac{1+1/3^2+1/5^2+1/7^2+...}{1-1/3^{p+1}+1/5^ {p+1}-1/7^{p+1}+...}.\] The constant is the best possible. We also provide related estimates for harmonic functions on Euclidean domains.

math.PR↗