Let $G$ be a real rank one connected semisimple Lie group with finite center. As well-known the radial, heat, and Poisson maximal operators satisfy the $L^p$-norm inequalities for any $p>1$ and a weak type $L^1$ estimate. The aim of this paper is to find a subspace of $L^1(G)$ from which they are bounded into $L^1(G)$. As an analogue of the atomic Hardy space on the real line, we introduce an atomic Hardy space on $G$ and prove that these maximal operators with suitable modifications are bounded from the atomic Hardy space on $G$ to $L^1(G)$.

We consider Lie groups whose Lie algebra is the nilradical of a parabolic subalgebra of a complex simple Lie algebra, endowed with left-invariant Hermitian metrics. For such Riemannian Lie groups, we describe the Lie algebras of their maximal isotropy groups.

If $P,Q:[0,\infty)\to$ are increasing functions and $T$ is the Calderón operator defined on positive or decreasing functions, then optimal modular inequalities $\int P(Tf)\leq C\int Q(f)$ are proved. If $P=Q$, the condition on $P$ is both necessary and sufficient for the modular inequality. In addition, we establish general interpolation theorems for modular spaces.

In this paper we consider the existence of periodic solutions of neutral functional differential equations. It has been proved that for convex neutral functional differential equations of $D$-operator type with finite (or infinite) delay and hyperneutral functional differential equations with finite delay, there is a periodic solution if and only if there is a bounded solution. The results proved by Massera, Chow and Makay are generalized.

Let $W$ be an irreducible subvariety of codimension $r$ in a smooth affine variety $X$ of dimension $n$ defined over the complex field $C$. Suppose that $W$ is left pointwise fixed by an automorphism of $X$ of infinite order or by a one-dimensional algebraic torus action on $X$. In the present article, we consider whether or not $X$ is then an affine space bundle over $W$ of fiber dimension $n-r$. Our results concern the case $r=1$ or the case $r=2$ and $n\leq3$. As by-products, we obtain algebro-topological characterizations of the affine 3-space.

In our preceding article, by applying the results of Bieberbach, Fatou and Picard, we have studied the domain of holomorphy of the solution of the Cauchy problem for the differential operator with coefficients of entire functions. In this article, by employing the results of the modular function and its ordinary differential equation, we give a remark on the domain of holomorphy of the solution of the Cauchy problem for the differential operator of principal part with polynomial coefficients.

Our primary object of this paper is to give a representation formula for a surface with prescribed mean curvature in the (metric) 3-sphere by means of a single component of the generalized Gauss map. For a CMC (constant mean curvature) surface, we derive another representation formula by means of the adjusted Gauss map. These formulas are spherical versions of the Kenmotsu representation formula for surfaces in the Euclidean 3-space. Spin versions of them are obtained as well.

A sufficient condition is obtained for the existence of a globally asymptotically stable positive almost periodic solution of a discrete logistic equation. The sufficient condition becomes a necessary and sufficient condition for the global asymptotic stability of the positive equilibrium of the corresponding autonomous equation.

Vogan-Zuckerman's standard representation $X$ for a real reductive group $G(R)$ is constructed from a $\theta$-stable parabolic subalgebra $\mathfrak{q}$ of the complexified Lie algebra $\mathfrak{g}$ of $G(R)$. Adams and Vogan showed that the set of $\mathfrak{g}$-principal $K$-orbits in the associated variety $\mathrm{Ass}(X)$ of $X$ is in one-to-one correspondence with the set $\mathcal{B}_{\mathfrak{g}^-}^L/K$ of $K$-conjugacy classes of $\theta$-stable Borel subalgebras of large type having representatives in the opposite parabolic subalgebra $\mathfrak{q}^-$ of $\mathfrak{q}$. In this paper, we give a description of $\mathcal{B}_{\mathfrak{q}}^L/K$ and show that $\mathcal{B}_{\mathfrak{q}}^L/K\ne\emptyset$ under certain condition on the positive system of imaginary roots contained in $\mathfrak{q}$. Furthermore, we construct a finite group which acts on $\mathcal{B}_{\mathfrak{q}}^L/K$ transitively.

Mapping properties for the homogeneous fractional integral operator $T_{{\mit \Omega},\alpha}$ on the Hardy spaces $H^p(R^n)$ are studied. Our results give the extension of Stein-Weiss and Taibleson-Weiss's results for the boundedness of the Riesz potential operator $I_{\alpha}$ on the Hardy spaces $H^p(R^n)$.