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.