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.

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 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.

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.

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.