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By introducing a class of new function spaces $B^{\sigma,s}_{p,q}$ as the resolution spaces, we study the Cauchy problem for the nonlinear Klein-Gordon equation (NLKG) in all spatial dimensions $d \geqslant 1$, $$ \partial^2_t u + u- \Delta u + u^{1+\alpha} =0, \ \ (u, \partial_t u)|_{t=0} = (u_0,u_1). $$ We consider the initial data $(u_0,u_1)$ in super-critical function spaces $E^{\sigma,s} \times E^{\sigma-1,s}$ for which their norms are defined by $$ \|f\|_{E^{\sigma,s}} = \|\langle\xi\rangle^\sigma 2^{s|\xi|}\widehat{f}(\xi)\|_{L^2}, \ s<0, \ \sigma \in \mathbb{R}. $$ Any Sobolev space $H^{\kappa}$ can be embedded into $ E^{\sigma,s}$, i.e., $H^\kappa \subset E^{\sigma,s}$ for any $ \kappa,\sigma \in \mathbb{R}$ and $s<0$. We show the global existence and uniqueness of the solutions of NLKG if the initial data belong to some $E^{\sigma,s} \times E^{ \sigma-1,s}$ ($s<0, \ \sigma \geqslant \max (d/2-2/\alpha, \, 1/2), \ \alpha\in \mathbb{N}, \ \alpha \geqslant 4/d$) and their Fourier transforms are supported in the first octant, the smallness conditions on the initial data in $E^{\sigma,s} \times E^{\sigma-1,s}$ are not required for the global solutions. Similar results hold for the sinh-Gordon equation $\partial^2_t u - \Delta u + \sinh u=0$ if the spatial dimensions $d \geqslant 2$., Comment: 57 pages

We consider the Cauchy problem for the complex valued semi-linear heat equation $$ \partial_t u - \Delta u - u^m =0, \ \ u (0,x) = u_0(x), $$ where $m\geq 2$ is an integer and the initial data belong to super-critical spaces $E^s_\sigma$ for which the norms are defined by $$ \|f\|_{E^s_\sigma} = \|\langle \xi\rangle^\sigma 2^{s|\xi|}\hat{f}(\xi)\|_{L^2}, \ \ \sigma \in \mathbb{R}, \ s<0. $$ If $s<0$, then any Sobolev space $H^{r}$ is a subspace of $E^s_\sigma$, i.e., $\cup_{r \in \mathbb{R}} H^r \subset E^s_\sigma$. We obtain the global existence and uniqueness of the solutions if the initial data belong to $E^s_\sigma$ ($s<0, \ \sigma \geq d/2-2/(m-1)$) and their Fourier transforms are supported in the first octant, the smallness conditions on the initial data in $E^s_\sigma$ are not required for the global solutions. Moreover, we show that the error between the solution $u$ and the iteration solution $u^{(j)}$ is $C^j/(j\,!)^2$. Similar results also hold if the nonlinearity $u^m$ is replaced by an exponential function $e^u-1$., Comment: 37 Pages

Modulation spaces $M^s_{p,q}$ were introduced by Feichtinger \cite{Fei83} in 1983. By resorting to the wavelet basis, B\'{e}nyi and Oh \cite{BeOh20} defined a modified version to Feichtinger's modulation spaces for which the symmetry scalings are emphasized for its possible applications in PDE. By carefully investigating the scaling properties of modulation spaces and their connections with the wavelet basis, we will introduce a class of generalized modulation spaces, which contain both Feichtinger's and B\'{e}nyi and Oh's modulation spaces. As their applications, we will give a local well-posedness and a (small data) global well-posedness results for NLS in some rougher generalized modulation spaces, which generalize the well posedness results of \cite{BeOk09} and \cite{WaHud07}, and certain super-critical initial data in $H^s$ or in $L^p$ are involved in these spaces., Comment: 40 pages, 6 figures

First, using the uniform decomposition in both physical and frequency spaces, we obtain an equivalent norm on modulation spaces. Secondly, we consider the Cauchy problem for the dissipative evolutionary pseudo-differential equation \partial_t u + A(x,D) u = F\big((\partial^\alpha_x u)_{|\alpha|\leq \kappa}\big), \ \ u(0,x)= u_0(x), where $A(x,D)$ is a dissipative pseudo-differential operator and $F(z)$ is a multi-polynomial. We will develop the uniform decomposition techniques in both physical and frequency spaces to study its local well posedness in modulation spaces $M^s_{p,q}$ and in Sobolev spaces $H^s$. Moreover, the local solution can be extended to a global one in $L^2$ and in $H^s$ ($s>\kappa+d/2$) for certain nonlinearities., Comment: 39 Pages

We will show its local well-posedness in modulation spaces $M^{1/2}_{2,q}({\Real})$ $(2\leq q<\infty) $. It is well-known that $H^{1/2}$ is a critical Sobolev space of DNLS so that it is locally well-posedness in $H^s$ for $s\geq 1/2$ and ill-posed in $H^{s'}$ with $s'<1/2.$ Noticing that that $M^{1/2}_{2,q} \subset B^{1/q}_{2,q}$ is a sharp embedding and $L^2 \subset B^0_{2,\infty}$, our result contains all of the subcritical data in $M^{1/2}_{2,q}$, which contains a class of functions in $L^2\setminus H^{1/2}$., Comment: 50 Pages

In this note we present the metric approximation property for weighted mixed-norm $L_w^{(p_1,\dots ,p_n)}$ and variable exponent Lebesgue type spaces. As a consequence, this also implies the same property for modulation and Wiener-Amalgam spaces. We then characterise nuclear operators on such spaces and state the corresponding Grothendieck-Lidskii trace formulae. We apply the obtained results to derive criteria for nuclearity and trace formulae for periodic operators on $\mathbb R^n$ and functions of the harmonic oscillator in terms of the global symbol., Comment: 11 pages. arXiv admin note: text overlap with arXiv:1410.4687

In this paper we first prove the metric approximation property for weighted mixed-norm $L_w^{(p_1,\dots ,p_n)}$ spaces. Using Gabor frame representation this implies that the same property holds in weighted modulation and Wiener amalgam spaces. As a consequence, Grothendieck's theory becomes applicable, and we give criteria for nuclearity and $r$-nuclearity for operators acting on these space as well as derive the corresponding trace formulae. Finally, we apply the notion of nuclearity to functions of the harmonic oscillator on modulation spaces., Comment: 22 pages; slightly updated final version, to appear in J. Lond. Math. Soc

We survey some recent progress on modulation spaces and the well-posedness results for a class of nonlinear evolution equations by using the frequency-uniform localization techniques., Comment: to appear in Evolution Equations of Hyperbolic and Schr\"odinger Type, Progress in Mathematics, Vol.301, Springer, 2012

First, we consider some fundamental properties including dual spaces, complex interpolations of $\alpha$-modulation spaces $M^{s,\alpha}_{p,q}$ with $0

We show the sharp global well posedness for the Cauchy problem for the cubic (quartic) non-elliptic derivative Schr\"odinger equations with small rough data in modulation spaces $M^s_{2,1}(\mathbb{R}^n)$ for $n\ge 3$ ($n= 2$). In 2D cubic case, using the Gabor frame, we get some time-global dispersive estimates for the Schr\"odinger semi-group in anisotropic Lebesgue spaces, which include a time-global maximal function estimate in the space $L^2_{x_1}L^\infty_{x_2,t}$. By resorting to the smooth effect estimate together with the dispersive estimates in anisotropic Lebesgue spaces, we show that the cubic hyperbolic derivative NLS in 2D has a unique global solution if the initial data in Feichtinger-Segal algebra or in weighted Sobolev spaces are sufficiently small., Comment: 42 Pages

Sufficient and necessary conditions for the generalized Gagliardo-Nirenberg (GN) inequality in Besov spaces and Triebel-Lizorkin spaces are obtained. Applying the GN inequality, we show that the finite-time blowup solutions have concentration phenomena in critical Lebesgue space L^3. Moreover, we consider the minimizer for a class of variational problem by applying the fractional GN inequality., Comment: 40 Pages

We prove the local well posedness for the KdV equation in the modulation space $M^{-1}_{2,1}(\mathbb{R})$. Our method is to substitute the dyadic decomposition by the uniform decomposition in the discrete Bourgain space. This wellposedness result enables us to show that the solution map is discontinuous at the origin with respect to the $H^{s} $-topology as soon as $s<-1$. Making use of the Miura transform we also deduce a discontinuity result for the $ H^s(\Real) $-topology, $s<0 $, for the solution map associated with the focussing and defocussing mKdV equations., Comment: 35 pages (improved version)

In this paper, we consider the trace theorem for modulation spaces, alpha modulation spaces and Besov spaces. For the modulation space, we obtain the sharp results., Comment: 26 pages