ALIBABA 2021 Problem 9

Let $\varepsilon$ be positive constant and $u$ satisfies that $$\begin{cases} (\partial_t-\varepsilon\partial_x^2-\partial_y^2)u=0, & (t,x,y) \in \mathbb{R}_+ \times \mathbb{R} \times \mathbb{R}_+, \partial_y u\vert_{y=0}=\partial_x h, &u\vert_{t=0}=0. & \end{cases}$$ Here $h(t,x)$ is a smooth Schwartz function. Define the operator $e^{a\langle D\rangle}$ $$\mathcal{F}_x(e^{a\langle D\rangle} f)(k)=e^{a\langle k\rangle} \mathcal{F}_x(f)(k), \langle k\rangle=1+\vert k\vert,$$ where $\mathcal{F}_x$ stands for the Fourier transform in $x$ . Show that $$\int_0^T \|e^{(1-s)\langle D\rangle} u\|_{L_{x,y}^2}^2 ds \le C \int_0^T \|e^{(1-s)\langle D\rangle} h\|_{H_x^{\frac{1}{4}}}^2 ds$$ with constant $C$ independent of $\varepsilon, T$ and $h$ .