Minxing Shen

Let $Δ_{\mathbb{T}^m\times \mathbb{R}^n}$ denote the Laplace-Beltrami operator on the product spaces $\mathbb{T}^m\times \mathbb{R}^n$. In this article we show that $$ \left\|e^{itΔ_{\mathbb{T}^m\times \mathbb{R}^n}}f\right\|_{L^p(\mathbb{T}^m\times \mathbb{R}^n\times [0,1])} \leq C \|f\|_{W^{α,p}(\mathbb{T}^m\times\mathbb{R}^n)} $$ holds if $p\geq 2(m+n+2)/(m+n)$ and $α> (m+2n)(1/2-1/p)-2/p$. Furthermore, we apply the $\ell^2$-decoupling inequalities to establish local $L^p$-smoothing estimates for the Schrödinger operator $e^{itΔ_{\mathbb{T}^m\times\mathbb{R}^n}}$ in modulation spaces $M_{p,q}^α(\mathbb{T}^m\times\mathbb{R}^n)$: $$ \|e^{itΔ_{\mathbb{T}^m\times\mathbb{R}^n}}f\|_{L^p(\mathbb{T}^m\times\mathbb{R}^n\times [0,1])}\leq C \|f\|_{M_{p,q}^α(\mathbb{T}^m\times\mathbb{R}^n)} $$ for some range of $α$ and $p, q$. The smoothing estimates in $L^p$-Sobolev and modulation spaces are sharp up to the endpoint regularity, in a certain range of $p$ and $q$.

We show that for an entire function $\varphi$ belonging to the Fock space ${\mathscr F}^2(\mathbb{C}^n)$ on the complex Euclidean space $\mathbb{C}^n$, the integral operator \begin{eqnarray*} S_{\varphi}F(z)=\int_{\mathbb{C}^n} F(w) e^{z \cdot\bar{w}} \varphi(z- \bar{w})\,d\lambda(w), \ \ \ \ \ z\in \mathbb{C}^n, \end{eqnarray*} is bounded on ${\mathscr F}^2(\mathbb{C}^n)$ if and only if there exists a function $m\in L^{\infty}(\mathbb{R}^n)$ such that $$ \varphi(z)=\int_{\mathbb{R}^n} m(x)e^{-2\left(x-\frac{i}{2} z \right)\cdot \left(x-\frac{i}{2} z \right)} dx, \ \ \ \ \ \ z\in \mathbb{C}^n. $$ Here $d\lambda(w)= \pi^{-n}e^{-\left\vert w\right\vert^2}dw$ is the Gaussian measure on $\mathbb C^n$. With this characterization we are able to obtain some fundamental results including the normaility, the algebraic property, spectrum and compactness of this operator $S_\varphi$. Moreover, we obtain the reducing subspaces of $S_{\varphi}$. In particular, in the case $n=1$, we give a complete solution to an open problem proposed by K. Zhu for the Fock space ${\mathscr F}^2(\mathbb{C})$ on the complex plane ${\mathbb C}$ (Integr. Equ. Oper. Theory {\bf 81} (2015), 451--454).