We investigate the stationary Navier-Stokes equations in Bessel-potential spaces with Muckenhoupt weights. Since in this setting it is possible that the solutions do not posses any weak derivatives, we use the notation of very weak solutions introduced by Amann [1]. The basic tool is complex interpolation, thus we give a characterization of the interpolation spaces of the spaces of data and solutions. Then we establish a theory of solutions to the Stokes equations in weighted Bessel-potential spaces and use this to prove solvability of the Navier-Stokes equations for small data by means of Banach's Fixed Point Theorem.

We study the asymptotic behavior in time of solutions to the Cauchy problem of nonlinear Schrödinger equations with a long-range dissipative nonlinearity given by $\lambda |u{|}^{p-1}u$ in one space dimension, where (namely, $p$ is a critical or subcritical exponent) and $\lambda$ is a complex constant satisfying Im and $((p-1)/2\sqrt{p})\left|Re\lambda \right|\le \left|Im\lambda \right|$ . We present the time decay estimates and the large-time asymptotics of the solution for arbitrarily large initial data, when “$p=3$ ” or “ and $p$ is suitably close to $3$ ”.

This article continues the asymptotic analysis of a nonlinear Schrödinger-Poisson system which models in a far from equilibrium regime the quantum transport in electronic devices like resonant tunneling diodes. Within the reduction to an $h$ -dependent linear problem with uniform regularity estimates for the potential already established in the first part, explicit computations of the asymptotic finite dimensional nonlinear system are derived. They rely on an accurate (phase-space) analysis of the tunnel effect which relies on some kind of Breit-Wigner formula and Fermi golden rule.

Let $X$ be a compact Kähler manifolds with a non-trivial holomorphic Poisson structure $\beta$ . Then there exist deformations $\{(J_{\beta t},{\psi }_t)\}$ of non-trivial generalized Kähler structures with one pure spinor on $X$ . We prove that every Poisson submanifold of $X$ is a generalized Kähler submanifold with respect to $(J_{\beta t},{\psi }_t)$ and provide non-trivial examples of generalized Kähler submanifolds arising as holomorphic Poisson submanifolds. We also obtain unobstructed deformations of bihermitian structures constructed from Poisson structures.

This paper is devoted to solving a degenerate parabolic integrodifferential equation with the Robin boundary condition. We begin with solving the equation without the integral delay term. For that purpose we introduce some new unknown function following Favini and Yagi [4] and construct the fundamental solution to the equation to be satisfied by it by the method of Kato and Tanabe [5]. Using this fundamental solution we transform the original problem to an easily solvable integral equation for the time derivative of the new unknown function.

We consider Schrödinger equations with variable coefficients, which are long-range type perturbations of the flat Laplacian on $R^n$ . We characterize the wave front set of solutions to Schrödinger equations in terms of the initial state. Then it is shown that the singularities propagates along the classical flow, and results are formulated in a semiclassical setting. Methods analogous to the long-range scattering theory, in particular a modified free propagator, are employed.

D. Luecking has recently proved that a Toeplitz operator with measure symbol on the Bergman space of unit disk has finite rank if and only if its symbols is a linear combination of point masses. In this note we extend this theorem to higher dimensional cases.

On the setting of the upper half-space $\boldsymbol{H}$ of the Euclidean $n$-spaces, we give a sharp norm esitame of the weighted harmonic Bergman projection on $L_\alpha^p$ for $1 < p < \infty$. Also, we obtain the norm estimate of the projection depending on $\alpha > -1$ when p is fixed.

This paper considers a class of continuous functions constructed as a series of iterates of the “tent map” multiplied by variable signs. This class includes Takagi's nowhere-differentiable function, and contains the functions studied by Hata and Yamaguti [Japan J. Appl. Math., 1 (1984), 183-199] and Kono [Acta Math. Hungar., 49 (1987), 315-324] as a proper subclass. A complete description is given of the differentiability properties of the functions in this class, and several statements are proved concerning their uniform and local moduli of continuity. The results are applied to generation of random functions.

A definition of Lévy processes with values in the space of probability measures was introduced by Shiga and Tanaka (Electronic J. Prob. 11 (2006)). It is shown that the Lévy process with values in the space of probability measures in law has a modification satisfying a certain condition. The modification is a Lévy process in the sense of Shiga and Tanaka. The Lévy-Itô decomposition of sample paths of the Lévy process satisfying the condition is derived.

We shall consider linear independence measures for the values of the functions $D_a(z)$ and $E_a(z)$ given below, which can be regarded as $q$-analogues of Euler's divergent series and the usual exponential series. For the $q$-exponential function $E_q(z)$, our main result (Theorem 1) asserts the linear independence (over any number field) of the values 1 and $E_q(\alpha_j)(j = 1,...,m)$ together with its measure having the exponent $\mu = O(m)$, which sharpens the known exponent $\mu = O(m^2)$ obtained by a certain refined version of Siegel's lemma (cf. [$\mathbf{1}$]). Let p be a prime number. Then Theorem 1 further implies the linear independence of the $p$- adic numbers $\prod_{n=0}^\infty(1 + kp^n), (k = 0,1,...,p-1)$, over $\boldsymbol{Q}$ with its measure having the exponent $\mu < 2p$. Our proof is based on a modification of Maier's method which allows to construct explicit Padé type approximations (of the second kind) for certain $q$-hypergeometric series.

It is well-known that there exist no homogeneous ruled real hypersurfaces in a complex projective space. On the contrary there exists the unique homogeneous ruled real hypersurface in a complex hyperbolic space. Moreover, it is minimal. We characterize geometrically this minimal homogeneous real hypersurface by properties of extrinsic shapes of some curves.