The Cauchy problem for a modified Zakharov system is proven to be locally well-posed for rough data in two and three space dimensions. In the three dimensional case the problem is globally well-posed for data with small energy. Under this assumption there also exists a global classical solution for sufficiently smooth data.

We shall give two criteria of Wiener type which characterize minimally thin sets and rarefied sets in a cylinder. We shall also show that a positive superharmonic function on a cylinder behaves regularly outside a rarefied set in a cylinder.

In this paper, we consider an initial value problem of the differential equation written in the title under an initial condition $x(T)=A$, $x'(T)=B$ $(0<T<\infty$, $0<A<\infty$, $-\infty <B<\infty)$. In the case $\lambda >0$, we conclude that if $T$ and $A$ are fixed arbitrarily, then there exists a number $B_{1}$ such that in every case of $B = B_{1}$, $B>B_{1}$, and $B<B_{1}$, we get analytical expressions of the solution of the initial value problem valid in the neighborhoods of the ends of the domain of the solution. Moreover we treat the case $\lambda <-1$. This case connects with the boundary layer theory of viscous fluids. The conclusions of this case are got directly from those of the case $\lambda >0$. Finally we discuss the case $T=0$ and $\lambda <-1$.

Let $\boldsymbol b^p_\alpha$ be the parabolic Bergman space, which is the Banach space of all $L^p$-solutions of the parabolic equation $(\partial/\partial t + (-\Delta)^{\alpha})u = 0$ on the upper half space $\boldsymbol R^{n+1}_+$ with $0 < \alpha \leq 1$. We discuss the relation of Toeplitz operators to Carleson measures.

In this paper, the authors study a class of multilinear singular integral operators on product of Hardy spaces. The boundedness of another class of multilinear singular integral operators is also discussed on product of Herz-type spaces. Moreover, as their special cases, the corresponding results of multilinear fractional integral operator and multilinear Calder\'on-Zygmund operator can be obtained, respectively.

In this paper we discuss the rigidity of the canonical isometric imbedding $\pmb{f}_0$ of the Hermitian symmetric space $Sp(n)/U(n)$ into the Lie algebra $\mathfrak{sp}(n)$. We will show that if $n \ge 2$, then $\pmb{f}_0$ is strongly rigid, i.e., for any isometric immersion $\pmb{f}_1$ of a connected open set $U$ of $Sp(n)/U(n)$ into $\mathfrak{sp}(n)$ there is a euclidean transformation $a$ of $\mathfrak{sp}(n)$ satisfying $\pmb{f}_1=a\pmb{f}_0$ on $U$.

The fundamental theorem (existence and uniqueness) for submanifolds of real space forms is well-known. We will discuss this theorem for some families of submanifolds in the framework of Grassmann geometries in a unified way. In particular, we show the fundamental theorem for half dimensional totally complex submanifolds of the quaternion projective space $\mathbb{H}P^n$ or the quaternion hyperbolic space $\mathbb{H}H^n$. This result is an affirmative answer to the conjecture by Alekseevsky and Marchiafava.