Mathematische Nachrichten最新文献

We obtain different results about the mean ergodicity of weighted composition operators when acting on the spaces $��H�\left(�B�\right)�$, $��H_b��\left(�B�\right)��$, and $��H^{\infty }��\left(�B�\right)��$, where $�B$ is the open unit ball of a Banach space, as well as about the compactness and the mean ergodicity of the multiplication operator. This study relates these properties of the operators with properties of the symbol and the weight defining such operators.

The paper is concerned with the completeness property of root functions of the Dirac operator with summable complex-valued potential and nonregular boundary conditions. We also obtain an explicit form for the fundamental solution system of the considered operator.

We consider the quasilinear diffusion problem of $��u�=�(�u^1�,�\text{…}�,�u^m�)�$

In this paper, we study the structural properties of Nevanlinna measures, that is, Borel measures that arise in the integral representation of Herglotz–Nevanlinna functions. In particular, we give a characterization of these measures in terms of their Fourier transform, characterize measures supported on hyperplanes including extremal measures, describe the structure of the singular part of the measures when some variable are set to a fixed value, and provide estimates for the measure of expanding and shrinking cubes. Corresponding results are stated also in the setting of the polydisc where applicable, and some of our proofs are actually performed via the polydisc.

In this paper, we study Lagrangian submanifolds of the pseudo-nearly Kähler $��\mathrm{SL}�(�2�,�R�)�\times �\mathrm{SL}�(�2�,�R�)�$. First, we show that they split into four different classes depending on their behavior with respect to a certain almost product structure on the ambient space. Then, we give a complete classification of totally geodesic Lagrangian submanifolds of this space.

A notable example due to Heier, Lu, Wong, and Zheng shows that there exist compact complex Kähler manifolds with ample canonical line bundle such that the holomorphic sectional curvature is negative semi-definite and vanishes along high-dimensional linear subspaces in every tangent space. The main result of this note is an upper bound for the dimensions of these subspaces. Due to the holomorphic sectional curvature being a real-valued bihomogeneous polynomial of bidegree (2,2) on every tangent space, the proof is based on making a connection with the work of D'Angelo on complex subvarieties of real algebraic varieties and the decomposition of polynomials into differences of squares. Our bound involves an invariant that we call the holomorphic sectional curvature square decomposition length, and our arguments work as long as the holomorphic sectional curvature is semi-definite, be it negative or positive.

We consider Campanato spaces with exponents $��\lambda �,�p�$ on domains of class $�C^{�0�,�\gamma �}$ in the N-dimensional Euclidean space endowed with a natural anisotropic metric depending on $�\gamma$. We discuss several results including the appropriate Campanato's embedding theorem and we prove that functions of those spaces can be extended to the whole of the Euclidean space without deterioration of the exponents $��\lambda �,�p�$.

In this work, we study the asymptotic behavior of the structurally damped wave equations arising from the viscoelastic mechanics. We are particularly interested in the complicated interaction of the time-dependent oscillating coefficients on the Dirichlet Laplacian operator and the structurally damped terms. On the one hand, by the application of WKB analysis, we explore the asymptotic energy estimates of the wave equations influenced by four types of oscillating mechanisms. On the other hand, in order to prove the optimality of the energy estimates for the critical cases, typical coefficients and initial Cauchy data will be constructed to show the lower bound of the energy growth rate by the application of instability arguments.

In 1980, Métivier characterized the analytic (and Gevrey) hypoellipticity of $�L^2$-solvable partial linear differential operators by a priori estimates. In this note, we extend this characterization to ultradifferentiable hypoellipticity with respect to Denjoy–Carleman classes given by suitable weight sequences. We also discuss the case when the solutions can be taken as hyperfunctions and present some applications.