In this survey we discuss the connection between commutator operators and functions of bounded mean oscillation, and at the same time outline the parallel story of the real Hardy space and weak factorization of these functions. We provide motivation as to why these questions are interesting and highlight the many different methods of proof that exist for proving results of the type discussed.

We give upper and lower bounds for the Hausdorff dimensions for a class of graph-directed measures when its underlying directed graph is the infinite \(N\)-ary tree. These measures are different from graph-directed self-similar measures driven by finite directed graphs and are not necessarily Gibbs measures. However our class contains several measures appearing in fractal geometry and functional equations, specifically, measures defined by restrictions of non-constant harmonic functions on the two-dimensional Sierpínski gasket, the Kusuoka energy measures on it, and, measures defined by solutions of de Rham’s functional equations driven by linear fractional transformations.

If \(\Gamma\) is the range of a Jordan curve that bounds a convex set in the plane, then \(\frac{1}{2}(\Gamma+\Gamma)=\mathsf{co}(\Gamma),\) where \(+\) is the Minkowski sum and \(\mathsf{co}\) is the convex hull. Answering a question of V. N. Ushakov, we construct a simple closed curve in \(\mathbb{R}^3\) whose range \(\Gamma\) satisfies \(\frac{1}{2}(\Gamma+\Gamma)=\mathsf{co}(\Gamma)=[0,1]^3.\) Also we show that such a simple closed curve cannot be rectifiable.

Let \(E\) be a subset of a doubling metric space \((X,d)\). We prove that for any \(s\in [0, \dim_{A}E]\), where \(\dim_{A}\) denotes the Assouad dimension, there exists a subset \(F\) of \(E\) such that \(\dim_{A}F=s\). We also show that the same statement holds for the lower dimension \(\dim_L\).

In this paper, we formulate an equivalent definition of the backwards Itô-Henstock integral of an operator-valued stochastic process with respect to a Hilbert space-valued \(Q\)-Wiener process using double Lusin condition. Moreover, we establish some versions of convergence theorems for this integral.

Like his colleagues de Prony, Petit, and Poisson at the Ecole Polytechnique, Cauchy used infinitesimals in the Leibniz-Euler tradition both in his research and teaching. Cauchy applied infinitesimals in an 1826 work in differential geometry where infinitesimals are used neither as variable quantities nor as sequences but rather as numbers. He also applied infinitesimals in an 1832 article on integral geometry, similarly as numbers. We explore these and other applications of Cauchy’s infinitesimals as used in his textbooks and research articles. An attentive reading of Cauchy’s work challenges received views on Cauchy’s role in the history of analysis and geometry. We demonstrate the viability of Cauchy’s infinitesimal techniques in fields as diverse as geometric probability, differential geometry, elasticity, Dirac delta functions, continuity and convergence.

We consider general formulations of the change of variable formula for Riemann integrals.

In the first part of this paper we investigate four types of convergence of sequences of functions in metric spaces which preserve continuity. Besides a consideration of locally, quasi and continuously uniform convergence, we introduce the notion of semi uniform convergence. We discuss how all types of convergence are related both to each other and to pointwise convergence, and illustrate their behavior by examples. Moreover, we show how some of the types of convergence can be used to characterize compactness of the domains the functions under consideration live in. In the second part we investigate sequences of composition operators in the space \(BV\) of functions of bounded variation in the sense of Jordan. We give criteria under which such sequences converge locally uniformly and semi uniformly and present a new and short proof for the fact that composition operators which map the space \(BV\) into itself are automatically continuous.

We provide an elementary approach to the development of the continuous functional calculus both for a single bounded normal operator, as well as for commuting tuples of bounded self-adjoint operators on a Hilbert space.