# 数学代写|偏微分方程代写partial difference equations代考|MATH4335

#### Doug I. Jones

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## 数学代写|偏微分方程代写partial difference equations代考|The classical FBI transform and its inversion

Following [Brós-Iagolnitzer, 1973], [Brós-Iagolnitzer, 1975] (see also [IagolnitzerStapp, 1972]) we introduce the following transform of an arbitrary function $f \in$ $L^1\left(\mathbb{R}^n\right)$ :
$$F f(x, \xi)=\int_{\mathbb{R}^n} \mathrm{e}^{i \xi \cdot(x-y)-|\xi | x-y|^2} f(y) \mathrm{d} y,$$
where $(x, \xi) \in \mathbb{R}^n \times \mathbb{R}^n$; this can be rewritten as the convolution
$$F f(x, \xi)=f(x) * \mathrm{e}^{i \xi \cdot x-|\xi | x|^2} .$$
The Lebesgue Dominated Convergence Theorem implies that $F f$ is a bounded and continuous function of $(x, \xi)$ in $\mathbb{R}^{2 n}$. The standard estimates for convolution further give
$$|\xi|^{\frac{1}{2} n} \int_{\mathbb{R}^n}|F f(x, \xi)| \mathrm{d} x \leq \pi^{\frac{1}{2} n}|f|_{L^1\left(\mathbb{R}^n\right)} .$$
We shall exploit the formula
$$\mathrm{D}x^\alpha \mathrm{D}{\xi}^\beta\left(\mathrm{e}^{i \xi \cdot(x-y)-|\xi | x-y|^2}\right)=P_{\alpha, \beta}(x-y, \xi) \mathrm{e}^{i \xi \cdot(x-y)-|\xi | x-y|^2},$$
where $P_{\alpha, \beta}(x, \xi)$ is a polynomial with respect to $x$ of degree $|\alpha+\beta|$ whose coefficients are smooth functions of $\xi$ in $\mathbb{R}^n \backslash{0}$ satisfying estimates of the kind $|f(\xi)| \lessgtr$ $|\xi|^{|\alpha+\beta|}+|\xi|^{1-|\beta|}$. Then, by making use of the estimate $|x|^m \leq C_m|\xi|^{-\frac{1}{2} m} \mathrm{e}^{-|\xi||x|^2}$ we conclude immediately that $F f(x, \xi)$ is a $C^{\infty}$ function of $(x, \xi) \in \mathbb{R}^n \times\left(\mathbb{R}^n \backslash{0}\right)$. And from the fact that $P_{\alpha, 0}(\xi, x-y)$ is a polynomial with respect to $(\xi,|\xi|(x-y))$ we conclude that $F f(x, \xi)$ is a $C^{\infty}$ function of $x \in \mathbb{R}^n$ for all $\xi \in \mathbb{R}^n$. The growth as $|\xi| \rightarrow+\infty$ of each partial derivative $\mathrm{D}_x^\alpha F f$ is at most polynomial.

The transformation $F$ can be extended to an arbitrary distribution $u \in \mathcal{D}{L^1}^{\prime}\left(\mathbb{R}^n\right)$, by which we mean a finite sum $u=\sum\alpha \mathrm{D}^\alpha f_\alpha$ of derivatives of $L^1$ functions $f_\alpha$ :
$$F u(x, \xi)=u(x) * \mathrm{e}^{i \xi \cdot x-|\xi||x|^2} .$$

## 数学代写|偏微分方程代写partial difference equations代考|The FBI transform and analyticity

In this section we show how to characterize the real-analyticity of a distribution by the exponential decay of its FBI transform (3.4.4) as $|\xi| \nearrow+\infty$. This palliates the unsatisfactory aspects of the Fourier transform as remarked following Proposition 2.1.3. We begin by proving a special case of (part of) the claim. As before $\Omega$ will be an open subset of $\mathbb{R}^n$.

Lemma 3.4.5 If $u \in \mathcal{E}^{\prime}(\Omega)$ and $u \equiv 0$ in a neighborhood of $x^{\circ}$ then the following is true:
(EXP DECAY 1) There is a neighborhood $U \subset \Omega$ of $x^{\circ}$ such that
$$\exists c>0, \forall(x, \xi) \in U \times \mathbb{R}^n,|F u(x, \xi)| \lesssim \mathrm{e}^{-c|\xi|} .$$
Proof If $u \equiv 0$ in a neighborhood of $x^{\circ}$ then $u$ admits a representation (1.3.5) with functions $f_\alpha \in L_{\mathrm{c}}^1\left(\mathbb{R}^n\right)$ which vanish identically in a neighborhood of $x^{\circ}$ : it suffices to select $\varepsilon<\operatorname{dist}\left(x^{\circ}, \operatorname{supp} u\right)$ in the construction leading to (1.3.5). As a consequence, it suffices to prove the claim for $u=\mathrm{D}x^\alpha f$ with $\alpha \in \mathbb{Z}{+}^n$ and $f \in L^1\left(\mathbb{R}^n\right), x^{\circ} \notin \operatorname{supp} f$. In this case, going back to (3.4.1) we get
\begin{aligned} F u(x, \xi) & =(-1)^{|\alpha|} \int_{\mathbb{R}^n} f(y) \mathrm{D}y^\alpha\left(\mathrm{e}^{i \xi \cdot(x y) \kappa|\xi||x y|^2}\right) \mathrm{d} y \ & =\int{\mathbb{R}^n} f(y) \mathrm{e}^{i \xi \cdot(x-y)-\kappa|\xi||x-y|^2} P_\alpha(|\xi|(x-y), \xi) \mathrm{d} y, \end{aligned}

where $P_\alpha(x, \xi)$ is a polynomial with respect to $(x, \xi)$ [cf. (3.4.3)]. We can use the inequalities $\left(|\xi|^{\frac{1}{2}}|x-y|\right)^m \leq C_m \kappa^{-\frac{1}{2} m} \mathrm{e}^{-\frac{1}{2} \kappa|\xi | x-y|^2}\left(m \in \mathbb{Z}{+}, \kappa>0\right)$ to obtain $$|F u(x, \xi)| \lesssim(1+|\xi|)^{|\alpha|} \int{\mathbb{R}^n}|f(y)| \mathrm{e}^{-\frac{1}{2} \kappa|\xi||x-y|^2} \mathrm{~d} y .$$

# 偏微分方程代写

## 数学代写|偏微分方程代写partial difference equations代考|The classical FBI transform and its inversion

$$F f(x, \xi)=\int_{\mathbb{R}^n} \mathrm{e}^{i \xi \cdot(x-y)-|\xi| x-\left.y\right|^2} f(y) \mathrm{d} y$$

$$F f(x, \xi)=f(x) * \mathrm{e}^{i \xi \cdot x-\left.|\xi| x\right|^2} .$$

$$|\xi|^{\frac{1}{2} n} \int_{\mathbb{R}^n}|F f(x, \xi)| \mathrm{d} x \leq \pi^{\frac{1}{2} n}|f|{L^1\left(\mathbb{R}^n\right)}$$ 我们将利用公式 $$\mathrm{D} x^\alpha \mathrm{D} \xi^\beta\left(\mathrm{e}^{i \xi \cdot(x-y)-|\xi| x-\left.y\right|^2}\right)=P{\alpha, \beta}(x-y, \xi) \mathrm{e}^{i \xi \cdot(x-y)}$$

$F f(x, \xi)$ 是一个 $C^{\infty}$ 的功能 $(x, \xi) \in \mathbb{R}^n \times\left(\mathbb{R}^n \backslash 0\right)$.

$(\xi,|\xi|(x-y))$ 我们的结论是 $F f(x, \xi)$ 是一个 $C^{\infty}$ 的功

$$F u(x, \xi)=u(x) * \mathrm{e}^{i \xi \cdot x-|\xi||x|^2} .$$

## 数学代写|偏微分方程代写partial difference equations代考|The FBI transform and analyticity

(EXP DECAY 1) 有一个邻域 $U \subset \Omega$ 的 $x^{\circ}$ 这样
$\exists c>0, \forall(x, \xi) \in U \times \mathbb{R}^n,|F u(x, \xi)| \lesssim \mathrm{e}^{-c|\xi|}$.

(1.3.5) $f_\alpha \in L_{\mathrm{c}}^1\left(\mathbb{R}^n\right.$ )在附近同样消失 $x^{\circ}$ : 选择就够了
$\varepsilon<\operatorname{dist}\left(x^{\circ}, \operatorname{supp} u\right)$ 在导致 (1.3.5) 的构造中。因 此，足以证明索赔 $u=\mathrm{D} x^\alpha f$ 和 $\alpha \in \mathbb{Z}+{ }^n$ 和 $f \in L^1\left(\mathbb{R}^n\right), x^{\circ} \notin \operatorname{supp} f$. 在这种情况下，回到 (3.4.1) 我们得到 $F u(x, \xi)=(-1)^{|\alpha|} \int_{\mathbb{R}^n} f(y) \mathrm{D} y^\alpha\left(\mathrm{e}^{i \xi \cdot(x y) \kappa|\xi||x y|^2}\right) \mathrm{d} y$ 在哪里 $P_\alpha(x, \xi)$ 是关于的多项式 $(x, \xi)$ [比昭。(3.4.3)]。 我们可以利用不等式 $\left(|\xi|^{\frac{1}{2}}|x-y|\right)^m \leq C_m \kappa^{-\frac{1}{2} m} \mathrm{e}^{-\frac{1}{2} \kappa|\xi| x-\left.y\right|^2}(m \in \mathbb{Z}+, \kappa>0)$ 获得
$$|F u(x, \xi)| \lesssim(1+|\xi|)^{|\alpha|} \int \mathbb{R}^n|f(y)| \mathrm{e}^{-\frac{1}{2} \kappa|\xi||x-y|^2} \mathrm{~d} y$$

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