Mathematische Nachrichten最新文献

The aim of this paper is to establish the boundedness of multiparameter singular integral operators associated with Zygmund dilations on product weighted Hardy spaces in the three-parameter setting. Additionally, we show that this class of operators are bounded on product Hardy spaces associated with ball quasi-Banach function spaces by employing the Rubio de Francia extrapolation technique. The generality of our result is illustrated by their applicability to concrete function spaces such as product Herz spaces and weighted product Morrey spaces. Even in these specific cases, the application yields entirely new results.

In this paper, we develop a new approach to investigation of the uniform stability for inverse spectral problems. We consider the non-self-adjoint Sturm–Liouville problem that consists in the recovery of the potential and the parameters of the boundary conditions from the eigenvalues and the generalized weight numbers. The special case of simple eigenvalues, as well as the general case with multiple eigenvalues, is studied. We find various subsets in the space of spectral data, on which the inverse mapping is Lipschitz continuous, and obtain the corresponding unconditional uniform stability estimates. Furthermore, the conditional uniform stability of the inverse problem under a priori restrictions on the potential is studied. In addition, we prove the uniform stability of the inverse problem by the Cauchy data, which are convenient for numerical reconstruction of the potential and for applications to partial inverse problems.

We study invariant Einstein metrics and Einstein–Randers metrics on the Stiefel manifold $��V_k�R^n�=�\mathrm{SO}��\left(�n�\right)��/�\mathrm{SO}��(�n�-�k�)��$. We use a characterization for (nonflat) homogeneous Einstein–Randers metrics as pairs of (nonflat) homogeneous Einstein metrics and invariant Killing vector fields. It is well known that, for Stiefel manifolds the isotropy representation contains equivalent summands, so a complete description of invariant metrics is difficult. We prove, by assuming additional symmetries, that the Stiefel manifolds $��V_{�1�+�k�}�R^{�1�+�2�k�}���(�k�>�2�)��$ and $��V_6�R^n���(�n�\ge �8�)��$

The trace-dev-div inequality in $�H^s$ controls the trace in the norm of $�H^s$ by that of the deviatoric part plus the $�H^{�s�-�1�}$ norm of the divergence of a quadratic tensor field different from the constant unit matrix. This is well known for $��s�=�0�$ and established for orders $��0�\le �s�\le �1�$ and arbitrary space dimension in this paper. For mixed and least-squares finite element error analysis in linear elasticity, this inequality allows to establish robustness with respect to the Lamé parameter $�\lambda$.

In this paper, we investigate the two-dimensional (2D), two-field magnetohydrodynamics (MHD) equations in the presence of a shear flow, assuming positive plasma viscosity and resistivity. We establish the global-in-time existence and uniqueness of a strong solution for the 2D two-field MHD equations under shearing-periodic boundary conditions, as proposed by Hawley et al. Moreover, we establish the existence and uniqueness of a strong solution for the linear advection-diffusion equation under shearing-periodic boundary condition by employing uniformly local $�L^2$ spaces.

In this paper, we study the nonlinear dissipative Boussinesq equation in the whole space $�R^n$ with $�L^1$ integrable data. As our preparations, the optimal estimates as well as the optimal leading terms for the linearized model are derived by performing the Wentzel–Kramers–Brillouin (WKB) analysis and the Fourier analysis. Then, under some conditions on the power $�p$ of nonlinearity, we demonstrate global (in time) existence of small data Sobolev solutions with different regularities to the nonlinear model by applying some fractional-order interpolations, where the optimal growth ($��n�=�2�$) and decay ($��n�⩾�3�$) estimates of solutions for large time are given. Simultaneously, we get a new large time asymptotic profile of global (in time) solutions. These results imply some influence of dispersion and dissipation on qualitative properties of solution.

We study the existence, uniqueness and polynomial stability of forward asymptotically almost periodic (AAP-) mild solutions for the wave equation with a singular potential on the whole space $�R^n$ in a framework of weak-$�L^p$ spaces. First, we use a Yamazaki-type estimate for wave groups on Lorentz spaces to establish the global well-posedness of bounded mild solutions for the corresponding linear wave equations. Then, we provide a Massera-type principle which guarantees the existence of AAP-mild solutions for linear wave equations. Using the results of linear wave equations and fixed point arguments we establish the well-posedness of such solutions for semilinear wave equations. Finally, we obtain a polynomial stability for mild solutions by employing dispersive estimates.

It is known that there exist complex solvmanifolds $��(�\Gamma �\setminus �G�,�J�)�$ whose canonical bundle is trivialized by a holomorphic section that is not invariant under the action of $�G$. The main goal of this paper is to classify the six-dimensional Lie algebras corresponding to such complex solvmanifolds, thus extending the previous work of Fino, Otal, and Ugarte for the invariant case. To achieve this, we complete the classification of six-dimensional solvable strongly unimodular Lie algebras admitting complex structures and identify among them, the ones admitting complex structures with Chern–Ricci flat metrics. Finally, we construct complex solvmanifolds with non-invariant holomorphic sections of their canonical bundle. In particular, we present an example of one such solvmanifold that is not biholomorphic to a complex solvmanifold with an invariant holomorphic section of its canonical bundle. Additionally, we discover a new six-dimensional solvable strongly unimodular Lie algebra equipped with a complex structure that has a nonzero holomorphic (3,0)-form.