Problem 1 (yau-contest 2024)
Section: Analysis and Differential Equations

Let $Q : \mathbb{R} \to \mathbb{R}$ be a $C_0^\infty$ function, i.e. it is smooth and has compact support. We assume $Q$ is even, i.e. $Q(x) = Q(-x)$. We assume $Q$ is non-trivial (i.e. $Q$ does not equal to zero everywhere).

Let $T_1(x) := xQ(x)$, and let $T_2(x) := x^2Q(x)$. Let $T_3 := e^{-x^2}(1 + x^{2024})$. We also introduce the following notation. For any $f : \mathbb{R} \to \mathbb{R}$, $\lambda > 0$, $\alpha \in \mathbb{R}$, we define

$$f_{\lambda,\alpha}(x) := \frac{1}{\lambda^{1/2}} f\!\left(\frac{x-\alpha}{\lambda}\right).$$

We claim: There exists $\delta > 0$, $\varepsilon > 0$, so that for any $c \in \mathbb{R}$ with $|c| < \delta$, one can find unique $\lambda, \alpha$ such that the followings hold

1. $|\lambda - 1| + |\alpha| < \varepsilon$.

2. $\langle Q_{\lambda,\alpha} - Q - cT_3,\ T_1 \rangle = 0$.

3. $\langle Q_{\lambda,\alpha} - Q - cT_3,\ T_2 \rangle = 0$.

(Here, for any two functions $f_1,f_2$, we define $\langle f_1,f_2\rangle := \int f_1(x)f_2(x)\,dx$.)

Is the above claim correct? Prove your conclusion.

Problem 2 (yau-contest 2024)
Section: Analysis and Differential Equations

Recall for every $f \in L^2(\mathbb{R}^3)$, one has that $g(x) := (-\Delta + 1)^{-1}f$ is a well-defined $L^2(\mathbb{R}^3)$ function. One may compute $g$ by solving

$$(-\Delta + 1)g = f. \tag{0.2}$$

(Recall $\Delta$ in $\mathbb{R}^3$ is defined as $\Delta = \sum_{i=1}^3 \partial_i^2$, also recall one may also define $(-\Delta + 1)^{-1}$ by Fourier theory.)

Now, let $V(x) := e^{-|x|^2}$, $x \in \mathbb{R}^3$. Prove that the operator $T := I + (-\Delta + 1)^{-1}V$ is invertible in $L^2$.

(Here, $Tf := f + (-\Delta + 1)^{-1}(Vf)$.)

Problem 3 (yau-contest 2024)
Section: Analysis and Differential Equations

Let $\psi(\xi) \in C_0^\infty(\mathbb{R})$ be smooth and has compact support. Let $\psi(\xi) = 0$, $\forall\,|\xi| \ge 1$. Let $f_1(\xi), f_2(\xi) \in C_0^\infty(\mathbb{R})$, i.e. $f_1,f_2$ are smooth and have compact support. Let $u_i : \mathbb{R} \times \mathbb{R} \to \mathbb{C}$, $i=1,2$, be defined as

$$u_1(x_1,x_2) := \int_{\mathbb{R}} \psi(\xi) f_1(\xi) e^{i\xi x_1} e^{i\xi^2 x_2} \, d\xi,$$

$$u_2(x_1,x_2) := \int_{\mathbb{R}} \psi(\eta - 10) f_2(\eta) e^{i\eta x_1} e^{i\eta^2 x_2} \, d\eta. \tag{0.3}$$

Prove there exists a constant $C$, which may depend on $\psi$, but does not depend on $f_1, f_2$, so that

$$\|u_1 u_2\|_{L^2(\mathbb{R}^2)} \le C\, \|f_1\|_{L^2(\mathbb{R})} \|f_2\|_{L^2(\mathbb{R})}. \tag{0.4}$$

(Hint: One may try to use Plancherel Theorem. It may be useful to observe that if one let $H(\xi,\eta) = f_1(\xi) f_2(\eta)$, then $\|H\|_{L^2(\mathbb{R}^2)}$ are also bounded by $\|f_1\|_{L^2(\mathbb{R})}\|f_2\|_{L^2(\mathbb{R})}$.)

Problem 4 (yau-contest 2024)
Section: Analysis and Differential Equations

Consider the heat equation in $\mathbb{R}^2$. Let $u = u(t,x)$ be a solution to

$$\begin{cases}
\dfrac{\partial u}{\partial t} - \Delta u = 0, \\
 u\big|_{t=0} = u_0 \in L^2.
\end{cases}$$

Then there exists a universal constant $C$ such that

$$\int_0^{\infty} \|u(t)\|_{L^2}^2 \, dt \le C\, \|u_0\|_{L^2}^2.$

Problem 5 (yau-contest 2024)
Section: Analysis and Differential Equations

Consider the Fourier transform. Let

$$Q(g,f)(x) := \int_{\mathbb{R}^N} \int_{\mathbb{S}^{N-1}} B\!\left(|x-y|, \frac{x-y}{|x-y|}\cdot\sigma\right)\, g(y)\, f(x')\, d\sigma\, d\theta\, dy,$$

where $B$ is a given two variable function, $\mathbb{S}^{N-1}$ stands for the unit sphere in $\mathbb{R}^N$ and

$$x' := \frac{x+y}{2} + \frac{|x-y|}{2}\,\sigma, \qquad y' := \frac{x+y}{2} - \frac{|x-y|}{2}\,\sigma.$$

Then

$$\widehat{Q}(g,f)(\xi) = (2\pi)^{-N/2} \int_{\mathbb{R}^N \times \mathbb{S}^{N-1}} \widehat{B}\!\left(|\eta|, \frac{\xi}{|\xi|}\cdot\sigma\right)\, \widehat{g}(\xi-\eta)\, \widehat{f}(\xi^{+}-\eta)\, d\sigma\, d\eta, \tag{0.5}$$

where $\widehat{B}(|\eta|,t) := \int_{\mathbb{R}} B(|q|,t) e^{-iq\eta}\, dq$, and $\xi^{\pm} := \dfrac{\xi \pm |\xi|\sigma}{2}$.