In this paper, we shall consider the class $N^p(D)(p>1)$ of holomorphic functions on the upper half plane $D:=\{ z \in {\bf C} \, | \, \verb|Im| z > 0 \}$ satisfying $\displaystyle \sup_{y>0} \int_{\bf R} \Bigl( \log (1+|f(x+iy)|) \Bigr)^p \,dx < \infty$. We shall prove that $N^p(D)$ is an $F$-algebra with respect to a natural metric on $N^p(D)$. Moreover, a canonical factorization theorem for $N^p(D)$ will be given.

We study propagation of microlocal regularities in the Sobolev space of solutions to boundary value problems for the isotropic elastic equation. We assume that the solutions microlocally belong to the Sobolev space of order s on the incident generalized bicharacteristic to the boundary. Then we discuss that whether the solutions have the same microlocal regularities in the Sobolev space on the reflected generalized bicharacteristic or not. Our results depend on the condition that how the incident generalized bicharacteristic attaches to the boundary. In this paper we only consider the boundary value problems for the isotropic elastic equation, however our method is valid for these of higher order hyperbolic equations and generalized elastic equations.

We complete the determination of the local Schur indices of each unipotent representation of the group $G(\mathbb F_q)$ of $\mathbb F_q$-rational points of a simple algebraic group $G$ defined over a finite field $\mathbb F_q$.

In this paper we introduce the sequence spaces $\ell _{\infty }(f,p,\sigma ,q,s),c_{0}(f,p,\sigma ,q,s)$ and $c(f,p,\sigma ,q,s)$ using a modulus function $f$ and defined over a seminormed space$(X,q)$, seminormed by $q$. We study some properties of these sequence spaces and obtain some inclusion relations.

In this paper, we show that if $Q\left( z\right) $ is a nonconstant polynomial, then every solution $w\not\equiv 0$ of the differential equation $w^{\left( n\right) }+e^{-z}w^{^{\prime }}+Q\left( z\right) w=0,$ has infinite order and we give an extension of this result. We will also show that if the equation $w^{\left( n\right) }+e^{-z}w^{^{\prime }}+cw=0$, where $c\neq 0$ is a complex constant, possesses a solution $w\not\equiv 0$ of finite order, then $c=-k^{n}$ where $% k$ is a positive integer. In the end, by study more general, we investigate the problem when $\sigma \left( Q\right) =1.$

It is first shown that a certain class of regular factorable matrices is equivalent to $C$, the Ces\'{a}ro matrix of order one. As a corollary it is shown that the matrix $A$ of [1] is equivalent to $C$. Then most of the results of [1] and [3] are extended to regular factorable matrices.

In this note, we study invariant subspaces of $L^2(\Bbb T^2)$ with respect to certain von Neumann algebras. We give a characterization of Beurling-type left-invariant subspaces of $L^2(\Bbb T^2)$. We also give a structure theorem of a non-trivial two-sided invariant subspace of $L^2(\Bbb T^2)$.

In this paper, for a hypersurface in the real space form of constant curvature, we prove that the unit normal vector field is an affine imbedding into a certain sphere bundle with canonical metric. Moreover, we study the relations between a hypersurface and its unit normal vector field as an affine imbedding. In particular, several hypersurfaces are characterized by affine geometric conditions which are independent of the choice of the transversal bundle.

Fig\`a-Talamanca characterized the space of Fourier multipliers as a dual space of a certain Banach space. In this paper, we give the similar result for bilinear Fourier multipliers.

If $x,y,z$ are real numbers satisfying $x+y+z=1$, then the maximum of the quadratic form $axy+bxz+cyz$ with positive constants $a,b,c$ is $$\displaystyle{\frac{abc}{2ab+2ac+2bc-a^2-b^2-c^2}}$$ under the assumption $\sqrt{a}<\sqrt{b}+\sqrt{c}$. Extending this fact, we give the maximum of the quadratic form $\displaystyle\sum_{1 \le i< j \le n} a_{ij}x_i x_j$ in $n$-variables $x_1,\ldots,x_n$ satisfying $\displaystyle\sum_{i=1}^{n} x_i = 1$ with constants $a_{ij} \ge 0$ under certain assumptions.

An inverse boundary value problem for the Helmholtz equation in a bounded domain is considered. The problem is to extract information about an unknown obstacle embedded in the domain with unknown impedance boundary condition (the Robin condition) from the associated Dirichlet-to-Neumann map. The main result is a characterization of the unknown obstacle via the sequences that are constructed by the Dirichletto- Neumann map, under smallness conditions on the wave number and the upper bound of the impedance. Moreover two alternative simple proofs of a previous result of Cheng- Liu-Nakamura which are based on only some energy estimates, an analysis of the blowup of the energy of so-called reflected solutions and an application of the enclosure method to the problem are also given.

We prove that $b$ is in $BMO(\R^n)$ if and only if the commutator $[b,I_{\alpha}]$ of the multiplication operator by $b$ and the fractional integral operator $I_{\alpha}$ is bounded from the classical Morrey space $L^{p,\lambda}(\R^n)$ to $L^{q,\mu}(\R^n)$, where $1<p< \infty$, $0<\alpha <n$, $0<\lambda <n- \alpha p$, $1/q=1/p-\alpha /n$ and $\lambda /p = \mu /q$. Also we will show that $b$ is in $\dot{\Lambda}_{\beta}(\R^n)$ if and only if the commutator $[b,I_{\alpha}]$ is bounded from the classical Morrey space $L^{p,\lambda}(\R^n)$ to $L^{q,\mu}(\R^n)$ or from $L^{p,\lambda}(\R^n)$ to $L^{q,\lambda}(\R^n)$, where $\alpha$ and $\beta$ satisfy some conditions.

We consider semilinear systems of wave equations with multiple propagation speeds and find out the critical order of the nonlinearity which characterizes large time behavior of small amplitude solutions to the system by establishing blowup results. We also evaluate the lifespan of the solution in terms of the size of the initial data from above and below. We underline that not only the order of the nonlinearity but also the way of coupling among unknowns in it has a major effect on the lifespan.