POTW Boundedness of Oscillatory Integrals

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Let $\phi \in C^\infty(\mathbb{R}^d \times \mathbb{R}^d)$ and $h\in C_0^\infty(\mathbb{R}^d \times \mathbb{R}^d)$ such that matrix $(\frac{\partial^2 \phi}{\partial x_j \, \partial y_k}(x,y))$ is invertible on the support of $h$. Show that for $1 \le p \le 2$, there is a constant $C = C_p > 0$ such that for every $\lambda > 0$ and $f\in L^p(\mathbb{R}^d)$, $$\left\|\int_{\mathbb{R}^d} e^{i\lambda\phi(x,y)}h(x,y)f(y)\, dy\right\|_{L_x^{q}(\mathbb{R}^d)} \le C\lambda^{-d/q}\|f\|_{L^p(\mathbb{R}^d)}$$ where $q$ is the conjugate exponent of $p$.

 

[wrobel]

Hormander's generalization of the Hausdorff-Young inequality
a tough task :)

 

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Spoiler

Let $$(T_\lambda f)(x) = \int_{\mathbb{R}^d} e^{i\lambda \phi(x,y)}\, h(x,y)f(y)\, dy$$ If $f\in L^1(\mathbb{R}^d)$, the triangle inequality gives an immediate estimate $\|T_\lambda f\|_{L^\infty(\mathbb{R}^d)} \lesssim\|f\|_{L^1(\mathbb{R}^d)}$. Now suppose $f\in L^2(\mathbb{R}^d)$. By Fubini's theorem we can write $$\|T_\lambda f\|_{L^2(\mathbb{R}^d)}^2 = \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \left(\int_{\mathbb{R}^d} e^{i\lambda[\phi(x,y) - \phi(x,z)]} h(x,y)\overline{h(x,z)}\, dx\right)\, f(y)\overline{f(z)}\, dy\, dz$$ For convenience, let the inner integral in paratheses be $K(y,z;\lambda)$. Since the matrix $D_xD_y\phi$ is invertible on the support of $h$, using a partition of unity if necessary we may assume $D_x[\phi(x,y) - \phi(x,z)] \ge M|y - z|$ on the supprt of $h$ for some $M > 0$. Then by method of stationary phase, $|K(y,z;\lambda)| \lesssim (1 + \lambda|y - z|)^{-N}$ for every positive integer $N$. If $N > n$, $\|(1 + \lambda |y|)^{-N}\|_{L^1(\mathbb{R}^d)} \simeq \lambda^{n-N}$; the Young and Schwarz inequalities produce estimates $$\|T_\lambda f\|_{L^2(\mathbb{R}^d)}^2 \lesssim \|(1 + \lambda|y|)^{-N}\|_{L^1(\mathbb{R}^d)} \|f\|_{L^2(\mathbb{R}^d)}^2 \lesssim \lambda^{n-N} \|f\|_{L^2(\mathbb{R}^d)}^2$$ Setting $N = 2n$ and taking square roots, we obtain $\|T_\lambda f\|_{L^2(\mathbb{R}^d)} \lesssim \lambda^{-n/2}\|f\|_{L^2(\mathbb{R}^d)}$. By Riesz interpolation of the linear operator $T_\lambda$ the result follows.