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The local and global well-posedness for the one dimensional fourth-order nonlinear Schr\"odinger equation are established in the modulation space $M^{s}_{2,q}$ for $s\geq \frac12$ and $2\leq q <\infty$. The local result is based on the $U^p-V^p$ spaces and crucial bilinear estimates. The key ingredient to obtain the global well-posedness is that we achieve a-priori estimates of the solution in modulation spaces by utilizing the power series expansion of the perturbation determinant introduced by Killip-Visan-Zhang for completely integrable PDEs., Comment: 41 pages

In this paper, we continue Curto-Hwang-Lee's work to study the connection between hyponormality and subnormality for block Toeplitz operators acting on the vector-valued Hardy space of the unit circle.Curto-Hwang-Lee's work focuses primarily on hyponormality and subnormality of block Toeplitz operators with rational symbols. By studying the "weak" commutativity, we extend Curto-Hwang-Lee's result to block Toeplitz operators with symbols of bounded type., Comment: arXiv admin note: text overlap with arXiv:1201.5976 by other authors

The paper is considered with the dissipative theory and feedback control under the framework of dissipation with the supply rate is the inner product of input u and the derivation of output y for the linear/nonlinear time-invariant input affine system. A necessary and sufficient condition of dissipativity of the class of systems is presented. Furthermore, we obtain necessary and sufficient conditions of the negative imaginary property, input strictly negative imaginary property and output strictly negative imaginary property for the nonlinear input affine system, which can cover the case that the linear time-invariant system in the literature. Besides, under given conditions, we consider the equivalence between the negative imaginary property and input-output stability. In other words, the system with negative imaginary property may be input-output stable under some certain conditions; conversely, the input-output stable system may also have negative imaginary property. Last but not least, we consider the equivalence between the asymptotical stability and strict dissipativity with the supply rate and some dissipative matrices are given for a certain linear time-invariant system.

We establish the sharp conditions for the embedding between Wiener amalgam spaces $W_{p,q}^s$ and some classical spaces, including Sobolev spaces $L^{s,r}$, local Hardy spaces $h_{r}$, Besov spaces $B_{p,q}^s$, which partially improve and extend the main result obtained by Guo et al. in J. Funct. Anal., 273(1):404-443, 2017. In addition, we give the full characterization of inclusion between Wiener amalgam spaces $W_{p,q}^s$ and $\alpha$-modulation spaces $M_{p,q}^{s,\alpha}$. Especially, in the case of $\alpha=0$ with $M_{p,q}^{s,\alpha} = M_{p,q}^s$, we give the sharp conditions of the most general case of these embedding. When $0

We study the Cauchy problem for the generalized KdV and one-dimensional fourth-order derivative nonlinear Schr\"odinger equations, for which the global well-posedness of solutions with the small rough data in certain scaling limit of modulation spaces is shown, which contain some data with infinite $L^{2}$ norm., Comment: 17 pages, all comments are welcome

We show some new local smoothing estimates of the fractional Schr\"odinger equations with initial data in $\alpha$-modulation spaces via decoupling inequalities. Furthermore, our necessary conditions show that the local smoothing estimates are sharp in some cases. As applications, the local smoothing estimates could show some new local well-posedness on modulation spaces of the fourth-order nonlinear Schr\"odinger equations on the line., Comment: 30 pages, 3 figures, all comments are welcome

We consider the Cauchy problem for the (derivative) nonlocal NLS in super-critical function spaces $E^s_\sigma$ for which the norms are defined by $$ \|f\|_{E^s_\sigma} = \|\langle\xi\rangle^\sigma 2^{s|\xi|}\hat{f}(\xi)\|_{L^2}, \ s<0, \ \sigma \in \mathbb{R}. $$ Any Sobolev space $H^{r}$ is a subspace of $E^s_\sigma$, i.e., $H^r \subset E^s_\sigma$ for any $ r,\sigma \in \mathbb{R}$ and $s<0$. Let $s<0$ and $\sigma>-1/2$ ($\sigma >0$) for the nonlocal NLS (for the nonlocal derivative NLS). We show the global existence and uniqueness of the solutions if the initial data belong to $E^s_\sigma$ and their Fourier transforms are supported in $(0, \infty)$, the smallness conditions on the initial data in $E^s_\sigma$ are not required for the global solutions., Comment: 36 Pages

In a $(1+n)$-dimensional Lorentz--Finsler manifold with $N$-Bakry--\'Emery Ricci curvature bounded below for $N\in(n,\infty]$, using the Riccati equation techniques, we established the Bishop--Gromov volume comparison for the so-called standard sets for comparisons in Lorentzian volumes (SCLVs).We also established the G\"unther volume comparison for SCLVs when the flag curvature was bounded above., Comment: arXiv admin note: text overlap with arXiv:1908.03832

We show an analogue of the Lorentzian splitting theorem for weighted Lorentz-Finsler manifolds: If a weighted Berwald spacetime of nonnegative weighted Ricci curvature satisfies certain completeness and metrizability conditions and includes a timelike straight line, then it necessarily admits a one-dimensional family of isometric translations generated by the gradient vector field of a Busemann function. Moreover, our formulation in terms of the $\epsilon$-range introduced in our previous work enables us to unify the previously known splitting theorems for weighted Lorentzian manifolds by Case and Woolgar-Wylie into a single framework., Comment: 28 pages, to appear in Internat. J. Math

The embedding relations between Besov-Triebel-Sobolev spaces and modulation spaces are determined explicitly. We extend the results of Sugimoto[2007]; Wang[2007] and Kobayashi[2011] to the most general cases. And we give the sharp embedding between Fourier $L^{p}$ spaces and modulation spaces., Comment: 19 pages, 2 figures

We establish the Bonnet-Myers theorem, Laplacian comparison theorem, and Bishop-Gromov volume comparison theorem for weighted Finsler manifolds as well as weighted Finsler spacetimes, of weighted Ricci curvature bounded below by using the weight function. These comparison theorems are formulated with $\epsilon$-range introduced in our previous paper, that provides a natural viewpoint of interpolating weighted Ricci curvature conditions of different effective dimensions. Some of our results are new even for weighted Riemannian manifolds and generalize comparison theorems of Wylie-Yeroshkin and Kuwae-Li., Comment: 39 pages; minor revisions, to appear in Anal. Geom. Metr. Spaces

We develop the theory of weighted Ricci curvature in a weighted Lorentz-Finsler framework and extend the classical singularity theorems of general relativity. In order to reach this result, we generalize the Jacobi, Riccati and Raychaudhuri equations to weighted Finsler spacetimes and study their implications for the existence of conjugate points along causal geodesics. We also show a weighted Lorentz-Finsler version of the Bonnet-Myers theorem based on a generalized Bishop inequality., Comment: 37 pages; some modifications to clarify motivation and improve presentation; to appear in J. Lond. Math. Soc