Conjugate $A$-harmonic tensors are generalizations of conjugate harmonic functions to differential forms. They share common analytical properties such as integrability and Holder continuity. Applications to quasiregular mappings follow.

For a Hermitian symmetric space $X = G/K$ of non-compact type let $\theta$ denote the Cartan involution of the semisimple Lie group $G$ with respect to the maximal compact subgroup $K$ of $G$, and let $q$ denote a $\theta$-stable parabolic subalgebra of the complexified Lie algebra $g$ of $G$ with corresponding Levi subgroup $L$ of $G$. Given a finite-dimensional irreducible $L$ module $F_{L}$ we find Bernstein-Gelfand-Gelfand type resolutions of the induced $(g, L \cap K)$ module $U(g) \otimes_{U(q)}F_{L}$ and its Hermitian dual, the produced module $\mathrm{Hom}_{U(\bar{q})}(U(g),F_{L})_{L \cap K-\mathrm{finite}}$, where $U(\cdot)$ is the universal enveloping algebra functor and $\bar{q}$ is the complex conjugate of $q$. The results coupled with a Grothendick spectral sequence provide for application to certain $(g,K)$ modules obtained by cohomological parabolic induction, and they extend results obtained initially by Stanke.

If $p = 3$ or 5 and $3 \leq n \lt p + 3$, then the homotopy type of $\mathbb{C}P^{n}$ contains only finitely many $PL$ homeomorphism types with a locally linear $PL$ action of the cyclic group of order $p$ fixing an isolated point.

In this paper we study the existence of reduced and irreducible complex or real projective curves contained in an ambient normal projective variety with prescribed singularities and with “low degree”. We consider germs of planar singularities of curves, up to topological or equisingular equivalence. The main result is an existence theorem for plane curves with ordinary singularities which improves previous results by Greuel, Lossen and Shustin.

A new interpolation problem is defined for the Bloch space and some partial results are obtained.

The main purpose of the paper is to establish formulae for both the upper and the lower multiplicity of an irreducible representation of a Moore group. Moreover, we show that for a group with small invariant neighbourhoods of the identity, the set of irreducible representations with finite upper multiplicity coincides with the dual of a quotient group.

Twisted transformation group $C^{\ast}$-algebras associated to locally compact dynamical systems $(X = Y/N,G)$ are studied, where $G$ is abelian, $N$ is a closed subgroup of $G$, and $Y$ is a locally trivial principal $G$-bundle over $Z = Y/G$. An explicit homomorphism between $H^{2}(G,C(X,\mathbb{T}))$ and the equivariant Brauer group of Crocker, Kumjian, Raeburn and Williams, $Br_{N}(Z)$, is constructed, and this homomorphism is used to give conditions under which a twisted transformation group $C^{\ast}$-algebra $C_{0}(X) \times_{\tau,\omega}G$ will be strongly Morita equivalent to another twisted transformation group $C^{\ast}$-algebra $C_{0}(Z) \times_{Id,\omega}N$. These results are applied to the study of twisted group $C^{\ast}$-algebras $C^{\ast}(\Gamma,\mu)$ where $\Gamma$ is a finitely generated torsion free two-step nilpotent group.

In this work we study some properties of the holomorphic TriebeI-Lizorkin spaces $H F^{pq}_{s}$, $0 \lt p$, $q \leq \infty$, $s \in \mathbb{R}$, in the unit ball $B$ of $\mathbb{C}^{n}$, motivated by some well-known properties of the Hardy-Sobolev spaces $H^{p}_{s} = H F^{p^{2}}_{s}$, $0 \lt p \lt \infty$.

We show that $\sum_{n \geq 0}|a_{n}|/(n + 1) \lesssim ||\sum_{n \geq 0}a_{n}z^{n}||_{H F^{1 \infty}_{0}}$, which improves the classical Hardy's inequality for holomorphic functions in the Hardy space $H^{1}$ in the disc. Moreover, we give a characterization of the dual of $HF^{1q}_{s}$, which includes the classical result $(H^{1})^{\ast} = \mathrm{BMOA}$. Finally, we prove some embeddings between holomorphic Triebel-Lizorkin and Besov spaces, and we apply them to obtain some trace theorems.

We consider the Dirichlet problem for the heat equation in domains with a minimally smooth, time-varying boundary. Our boundary data is taken to belong to a parabolic Sobolev space having a tangential (spatial) gradient, and $1/2$ of a time derivative, in $L^{p}$, $1 \lt p \lt 2 + \epsilon$. We obtain sharp $L^{p}$ estimates for the parabolic non-tangential maximal function of the gradient of our solutions.

We analyze bilipschitz group actions on Jordan curves and present a list of alternative descriptions for bilipschitz homogeneous bounded turning Jordan curves in doubling metric spaces.

The Whitehead asphericity conjecture claims that if $\langle\mathcal{A}||\mathcal{R}\rangle$ is an aspherical group presentation, then for every $\mathcal{S} \subset \mathcal{R}$ the subpresentation $\langle\mathcal{A}||\mathcal{S}\rangle$ is also aspherical. It is proven that if the Whitehead conjecture is false then there is an aspherical presentation $E = \langle\mathcal{A}||\mathcal{R} \cup \{z\}\rangle$ of the trivial group $E$, where the alphabet $\mathcal{A}$ is finite or countably infinite and $z \in \mathcal{A}$, such that its subpresentation $\langle\mathcal{A}||\mathcal{R}\rangle$ is not aspherical. It is also proven that if the Whitehead conjecture fails for finite presentations (i.e., with finite $\mathcal{A}$ and $\mathcal{R}$) then there is a finite aspherical presentation $\langle\mathcal{A}||\mathcal{R}\rangle$, $\mathcal{R} = \{R_{1},R_{2},\ldots,R_{n}}$, such that for every $\mathcal{S} \subseteq \mathcal{R}$ the subpresentation $\langle\mathcal{A}||\mathcal{S}\rangle$ is aspherical and the subpresentation $\langle\mathcal{A}||R_{1}R_{2},R_{3},\ldots,R_{n}$ of aspherical $\langle\mathcal{A}||R_{1}R_{2},R_{3},\ldots,R_{n}$ is not aspherical.