Paul A. Hagelstein

Tauberian conditions, Muckenhoupt weights, and differentiation properties of weighted bases

Let B be a homothecy invariant collection of convex sets in R. Given a measure μ, the associated weighted geometric maximal operator MB,μ is defined by MB,μf(x) ∶= sup x∈B∈B 1

On the stability of cycles by delayed feedback control

We present a delayed feedback control (DFC) mechanism for stabilizing cycles of one-dimensional discrete time systems. In particular, we consider a DFC for stabilizing -cycles of a differentiable function of the form where with . Following an approach of Morgül, we construct a map whose fixed points correspond to -cycles of . We then analyse the local stability of the above DFC mechanism by evaluating the stability of the corresponding equilibrium points of . We associate to each periodic orbit of an explicit polynomial whose Schur stability corresponds to the stability of the DFC on that orbit. This polynomial is the characteristic polynomial of a Jacobian matrix that lies in a large class of matrices that encompasses the usual ‘companion matrices’ found in linear algebra; the primary purpose of this paper is to show that this polynomial may be expressed in a surprisingly simple form. An example indicating the efficacy of this method is provided.

Weighted Solyanik Estimates for the Hardy–Littlewood Maximal Operator and Embedding of into

Let denote a weight in which belongs to the Muckenhoupt class and let denote the uncentered Hardy–Littlewood maximal operator defined with respect to the measure . The sharp Tauberian constant of with respect to , denoted by , is defined by In this paper, we show that the Solyanik estimate

On restricted weak type (1,1): The continuous case

\begin{aligned} \lim _{\alpha \rightarrow 1^-}\mathsf{C}_{w}(\alpha ) = 1 \end{aligned}

Solyanik estimates and local Hölder continuity of halo functions of geometric maximal operators

limα→1-Cw(α)=1holds. Following the classical theme of weighted norm inequalities we also consider the sharp Tauberian constants defined with respect to the usual uncentered Hardy–Littlewood maximal operator and a weight : We show that we have if and only if . As a corollary of our methods we obtain a quantitative embedding of into .

Maximal Operators Associated to Sets of Directions of Hausdorff and Minkowski Dimension Zero

Let \(\mathcal{B}\) denote a collection of open bounded sets in \(\mathbb{R}^{n}\), and define the associated maximal operator \(M_{\mathcal{B}}\) by

Sharp inequalities for one-sided Muckenhoupt weights

\displaystyle{M_{\mathcal{B}}f(x)\,:=\,\sup _{x\in R\in \mathcal{B}} \frac{1} {\vert R\vert }\int _{R}\vert f\vert.}