In this paper we completely classify parallel non-degenerate surfaces in 4-dimensional Lorentzian space forms. In addition, we also completely classify non-degenerate surfaces with parallel mean curvature vector in 4-dimensional Lorentzian space forms.

We prove the $L^p (1<p<\infty)$ estimates for the singular integrals with rough variable kernels. The $L^p$ boundedness of a class of modified directional Hilbert transforms is also given. As a consequence of this result, we get a good estimate for the singular integrals with rough odd kernels.

In this paper, we investigate relation between pseudo-Hermitian symmetric pairs and para-Hermitian symmetric ones.

We study arithmetic varieties $V$ attached to certain inner forms of $\boldsymbol{Q}$-rank one of the split symplectic $\boldsymbol{Q}$-group of degree two. These naturally arise as unitary groups of a 2-dimensional non-degenerate Hermitian space over an indefinite rational quaternion division algebra. First, we analyze the canonical mixed Hodge structure on the cohomology of these quasi-projective varieties and determine the successive quotients of the corresponding weight filtration. Second, by interpreting the cohomology groups within the framework of the theory of automorphic forms, we determine the internal structure of the cohomology “at infinity” of $V$, that is, the part which is spanned by regular values of suitable Eisenstein series or residues of such. In conclusion, we discuss some relations between the mixed Hodge structure and the so called Eisenstein cohomology. For example, we show that the Eisenstein cohomology in degree two consists of algebraic cycles.

We consider the real rank one unitary group $G$ and its subgroup $H$ obtained as the stabilizer of an anisotropic vector in the skew-hermitian space defining $G$. We compute the inner-product of an Eisenstein series on $H$ and a non-holomorphic cuspidal Hecke eigenform on $G$ restricted to $H$ to obtain an integral representation of the standard $L$-function of the eigenform. We also discuss some consequences of the integral representation.