We study generic and conformally flat hypersurfaces in Euclidean four-space. What kind of conformally flat three manifolds are really immersed generically and conformally in Euclidean space as hypersurfaces? According to the theorem due to Cartan [1], there exists an orthogonal curvature-line coordinate system at each point of such hypersurfaces. This fact is the first step of our study. We classify such hypersurfaces in terms of the first fundamental form. In this paper, we consider hypersurfaces with the first fundamental forms of certain specific types. Then, we give a precise representation of the first and the second fundamental forms of such hypersurfaces, and give exact shapes in Euclidean space of them.

We give a detailed survey, known and new results on the domain and the range of the maximal operator. In particular we employ the grand $L^{p}$ spaces and logarithmic Lebesgue spaces.

An adjoint variety $X(\mathfrak{g})$ associated to a complex simple Lie algebra $\mathfrak{g}$ is by definition a projective variety in $\mathbb {P}_{*}(\mathfrak{g})$ obtained as the projectivization of the (unique) non-zero, minimal nilpotent orbit in $\mathfrak{g}$. We first describe the tangent loci of $X(\mathfrak{g})$ in terms of $\mathfrak{sl}_{2}$-triples. Secondly for a graded decomposition of contact type $\mathfrak{g} = \oplus_{-2 \le i \le 2}\, \mathfrak{g}_{i}$, we show that the intersection of $X(\mathfrak{g})$ and the linear subspace $\mathbb {P}_*(\mathfrak{g}_{1})$ in $\mathbb {P}_{*}$$(\mathfrak{g})$ coincides with the cubic Veronese variety associated to $\mathfrak{g}$.

In this paper, we obtain some new existence theorems of the maximal and minimal fixed points for discontinuous increasing operators in $C[I, E]$, where $E$ is a Banach space. As applications, we consider the maximal and minimal solutions of nonlinear integro-differential equations with discontinuous terms in Banach spaces.

It is shown that any compact Levi flat manifold admitting a positive line bundle is embeddable into $\mathbb{P}^{n}$ by a CR mapping with an arbitrarily high, though finite, order of regularity.

A real hypersurface $M$ in a complex manifold $X$ is said to be Levi flat if it separates $X$ locally into two Stein pieces. It is proved that there exist no real analytic Levi flat hypersurfaces in ${\Bbb P}^2$.

For every $r$-th order Weil functor $T^{A}$, we introduce the underlying $k$-th order Weil functors $T^{A_{k}}$, $k = 1, \ldots, r-1$. We deduce that $T^{A}M \to T^{A_{r-1}} M$ is an affine bundle for every manifold $M$. Generalizing the classical concept of contact element by C. Ehresmann, we define the bundle $\mathcal{k}T^{A}M$ of contact elements of type $A$ on $M$ and we describe some affine properties of this bundle.

We characterize those positive measure $\mu$'s on the higher dimensional unit ball such that "two-weighted inequalities" hold for holomorphic functions and their derivatives. Characterizations are given in terms of the Carleson measure conditions. The results of this paper also distinguish between the fractional and the tangential derivatives.

Let $D$ be a domain in ${\Bbb R}^{m+1}$ and $E$ be a domain in ${\Bbb R}^{n+1}$. A pair of a smooth mapping $f:D \to E$ and a smooth positive function $\varphi$ on $D$ is called a caloric morphism if $\varphi\cdot u \circ f$ is a solution of the heat equation in $D$ whenever $u$ is a solution of the heat equation in $E$. We give the characterization of caloric morphisms, and then give the determination of caloric morphisms. In the case of $m<n$, there are no caloric morphisms. In the case of $m=n$, caloric morphisms are generated by the dilation, the rotation, the translation and the Appell transformation. In the case of $m>n$, under some assumption on $f$, every caloric morphism is obtained by composing a projection with a direct sum of caloric morphisms of ${\Bbb R}^{n+1}$.

Let $k$ be a real quadratic field and $\mathfrak{o}_{k}$, $E$ the ring of integers and the group of units in $k$. Denoting by $E(\mathfrak{p})$ the subgroup represented by $E$ of $(\mathfrak{o}_{k}/\mathfrak{p})^{\times}$ for a prime ideal $\mathfrak{p}$, we show that prime ideals $\mathfrak{p}$ for which the order of $E(\mathfrak{p})$ is theoretically maximal have a positive density under the Generalized Riemann Hypothesis.