We deal with the Lee conjecture (compact strictly pseudoconvex CR manifolds whose CR structure has a vanishing first Chern class admit a global pseudo-Einstein structure). We solve in affirmative the Lee conjecture for compact strictly pseudoconvex CR manifolds with a regular (in the sense of R. Palais, [Pal]) contact vector. The regularity assumption leads (via the Boothby-Wang theorem ([BoO-Wan]) and B. O’Neill’s f- damental equations of a submersion ([Nei])) to zero pseudohermitian torsion (and we may apply a result of [Lee2]). Moreover we construct a family ${\mathbf H}_{n}(s)$, $0<s<1$, of compact strictly pseudoconvex CR manifolds, so that each ${\mathbf H}_{n}(s)$ satisfies the Lee conjecture. We endow ${\mathbf H}_{n}(s)$ with the contact form (4); our construction is reminiscent of W. C. Boothby’s Hermitian metric (cf. [Boo]) on a complex Hopf manifold.

In this paper we characterize bounded subsets of the spaces $\mathfrak{D}_{L^{p}}'$, $1\leq p \leq\infty$, of distributions of $L^{p}$ growth. Moreover, we give necessary and sufficient conditions on a sequence in $\mathfrak{D}_{L^{p}}'$ to converge to 0.

Let $w$ be a weight function on a locally compact group $G$, so that the weighted $L_{1}$-space $L_{1}(w)$ forms a Banach algebra under convolution. Suppose that $G$ acts on a locally compact space $\Omega$, and that $B$ is a Banach space consisting of Radon measures on $\Omega$ which is also a left Banach $L_{1}(w)$-module. Under certain conditions on $B$, we shall characterize those bounded linear operators $T:L_{1}(w)\arrow B$ which satisfy $T(f*g)=f*T(g)$. We shall also show that there are numerous examples of Lorentz spaces which form left Banach $L_{1}(w)$-modules with respect to appropriate weight functions.

We study the existence of positive radial solutions of $(-1)^{m} \Delta^{m}u=g(|x|)f(u)$ in an annulus with Dirichlet boundary conditions. In particular $L^{\infty}$ a priori bounds are obtained.

We generalize the join of hypergroups as follows: If $H$ is an open subhypergroup of a hypergroup $K$ and $W$ a compact subhypergroup of a hypergroup $L$ such that $L/W=H$, then there is a natural hypergroup structure on the disjoint union $M:=(K-H)\cup L$. Properties of this hypergroup $M$ are discussed, and its Haar measure and its dual space are determined. As an application we determine the conjugacy class hypergroups $G^{G}$ as well as the dual hypergroups $\hat{G}$ of some compact groups $G$ which are close to the commutative case.