Rami Ayoush's research while affiliated with University of Warsaw and other places

We prove that the polar decomposition of the singular part of a vector measure depends on its conditional expectations computed with respect to the $q$-regular filtration. This dependency is governed by a martingale analog of the so-called wave cone, which naturally corresponds to the result of De Philippis and Rindler about fine properties of PDE-constrained vector measures. As a corollary we obtain a martingale version of Alberti's rank-one theorem.

We establish various forms of the following certainty principle: a set S⊂Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S \subset {\mathbb {R}}^{n}$$\end{document} contains a given finite linear pattern if S is a support of the Fourier transform of a sufficiently singular probability measure on Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^{n}$$\end{document}. As its main corollary, we provide new dimensional estimates for PDE- and Fourier-constrained vector measures. Those results, in certain cases of restrictions given by homogeneous operators, improve known bounds related to the notion of the k-wave cone.

In this article we provide lower bounds for the lower Hausdorff dimension of finite measures assuming certain restrictions on their quaternionic spherical harmonics expansion. This estimate is an analog of a result previously obtained by the authors for the complex spheres.

In this paper we study the dependence of geometric properties of Radon measures, such as Hausdorff dimension and rectifiability of singular sets, on the wavefront set. We prove our results by adapting the method of Brummelhuis to the non-analytic case. As an application we obtain a general form of the Uncertainty Principle for measures on the complex sphere. This instance of the UP generalizes the celebrated theorem of Aleksandrov and Forelli concerning regularity of pluriharmonic measures.

We establish various forms of the following certainty principle: a set $S \subset \mathbb{R}^{n}$ contains a given finite linear pattern, provided that $S$ is a support of the Fourier transform of a sufficiently singular probability measure on $\mathbb{R}^{n}$. As its main corollary, we provide new dimensional estimates for PDE- and Fourier-constrained vector measures. Those results, in certain cases of restrictions given by homogeneous operators, improve known bounds related to the notion of the $k$-wave cone.

We provide an estimate from below for the lower Hausdorff dimension of measures on the unit circle based on the arithmetic properties of their spectra. We obtain those bounds via adaptation of our previous results for martingales on \(q\)-regular trees to a specific backwards martingale. To show the sharpness of our method, we improve the best numerical lower bound known for the Hausdorff dimension of certain Riesz products.

In this paper we study the dependence of geometric properties of Radon measures, such as Hausdorff dimension and rectifiability of singular sets, on the wavefront set. This is achieved by adapting the method of Brummelhuis to the non-analytic case. As an application we obtain a general form of uncertainty principle for measures on the complex sphere which subsumes certain classical results about pluriharmonic measures.