In this paper, we investigate fine properties of measures belonging to a certain class of vector-valued measures arising as a natural generalization of gradients of BV functions. Our results concern dimensional estimates and rectifiability. We propose two conditions on the Fourier transform under which such measures cannot be very singular. The first one is related to the Fourier antisymmetry. The second one is a modification of the 2-wave cone condition for $\mathcal{A}$-free measures.

We study the notions of nuclearity and exactness for module maps on $C^{\ast }$-algebras which are $C^{\ast }$-module over another $C^{\ast }$-algebra with compatible actions and examine finite approximation properties of such $C^{\ast }$-modules. We prove module versions of the results of Kirchberg and Choi–Effros. As a concrete example, we extend the finite dimensional approximation properties of reduced $C^{\ast }$-algebras and von Neumann algebras on countable discrete groups to these operator algebras on countable inverse semigroups with the module structure coming from the action of the $C^{\ast }$-algebras on the subsemigroup of idempotents.

For Kähler manifolds, we explicitly determine the solution to the conformal Killing form equation in middle degree. In particular, we complete the classification of conformal Killing forms on compact Kähler manifolds. We give the first examples of conformal Killing forms on Kähler manifolds not coming from Hamiltonian 2-forms. These are supported by Calabi-type manifolds over a Kähler Einstein base. In this setup, we also give structure results and examples for the closely related class of Hermitian Killing forms.

We prove a transversality “lifting property” for compactified configuration spaces as an application of the multijet transversality theorem: given a submanifold of configurations of points on an embedding of a compact manifold M in Euclidean space, we can find a dense set of smooth embeddings of M for which the corresponding configuration space of points is transverse to any submanifold of the configuration space of points in Euclidean space, as long as the two submanifolds of compactified configuration space are boundary-disjoint. We use this setup to provide an attractive proof of the square-peg problem: there is a dense family of smoothly embedded circles in the plane where each simple closed curve has an odd number of inscribed squares, and there is a dense family of smoothly embedded circles in ${\mathbb{R}}^n$ where each simple closed curve has an odd number of inscribed square-like quadrilaterals.

We study delta-points in Banach spaces $h_{\mathcal{A},p}$ generated by adequate families $\mathcal{A}$, where $1\le p<\mathrm{\infty }$. When $p>1$, we prove that neither $h_{\mathcal{A},p}$ nor its dual contain delta-points. Under the extra assumption that $\mathcal{A}$ is regular, we prove that the same is true when $p=1$. In particular, the Schreier spaces and their duals fail to have delta-points. If $\mathcal{A}$ consists only of finite sets, we are able to rule out the existence of delta-points in $h_{\mathcal{A},1}$ and Daugavet-points in its dual.

We also show that if $h_{\mathcal{A},1}$ is polyhedral, then it is either (I)-polyhedral or (V)-polyhedral (in the sense of Fonf and Veselý).

We use embeddings of Bergman spaces and fractional radial differential operators to study the $L_{\mathit{\alpha }}^p$–$L_{\mathit{\beta }}^q$ mapping properties of Bergman-type projections on the open unit ball of ${\mathbb{C}}^n$. We recover and extend several existing results in this direction. In particular, this approach allows us to treat the cases when $0<q<1$ and $\mathit{\alpha }\le -1$. We also characterize when the embedding of one weighted Bergman space into another is compact.

With the use of real-variable techniques, we construct a weight function ω on the interval $[0,2\mathrm{\pi })$ which is doubling and satisfies $log\mathit{\omega }$ is a BMO function, but which is not a Muckenhoupt weight ($A_{\mathrm{\infty }}$). Applications to the BMO–Teichmüller space and the space of chord-arc curves are considered.