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Redundancy. Let us start with the Fourier transformation: a fonction $f$, say of one real variable (which could be only a tempered distribution) can be written as $$ f(x)=\int_{\mathbb R}\hat f(\xi) e …

On the Wikipedia page, one can read that an iff condition for L1 boundedness of the Fourier multiplier m(D) is that $$ \hat m\quad\text{ is a Borel measure with finite total mass. } $$ There is no ref …

Let us consider in one dimension the Fourier multiplier $\vert D\vert$ and the derivative $iD$. Both are well-defined on the Schwartz space $\mathscr S(\mathbb R)$ with the derivative sending $\mathsc …

Consider the polynomial $ P(\xi)=\prod_{1\le j\le n}(\xi-\lambda_j). $ The inverse Fourier transform of $(\xi-\lambda_j)$ is $$ \int(\xi-\lambda_j) e^{2iπ x\xi} d\xi=(D_x-\lambda_j)(\delta_0)=\frac{\ …

Let me change your notations slightly: you work in three dimensions and you want to compute $$ u_{jk}(x)=\int e^{2i\pi x\cdot \xi} \frac{\xi_j\xi_k}{\vert \xi\vert^4} d\xi, $$ where the integral shoul …

I would like to calculate "explicitly" the following integral, which is a Fourier transform: let $\alpha>0$ be a parameter, for $x\in \mathbb R$, we define $$ I(\alpha, x)=\int_\mathbb R \cos(xt) e^{- …

Let $\mu$ be a Radon measure on $\mathbb R^d$ with finite total mass: I guess that it is a tempered distribution on $\mathbb R^d$ and thus one may consider its Fourier transform. Now I guess that its …

To make sense of your Fourier multiplier, you need only to assume that $p\mapsto G(\vert p\vert^2)$ is a temperate distribution on $\mathbb R^d$. It is true whenever $G$ is a continuous function incre …

Let me first recall a particular case of the classical Hörmander-Mikhlin multiplier theorem: Let $m$ be a bounded function on $\mathbb {R} ^{n}$ which is smooth except possibly at the origin, and suc …

The Fourier transform $\hat u$ is defined on the Schwartz space $\mathscr S(\mathbb R^n)$ by $ \hat u(\xi)=\int e^{-2iπ x\cdot \xi} u(x) dx. $ It is an isomorphism of $\mathscr S(\mathbb R^n)$ and the …

Abstract nonsense: the Fourier inversion formula is valid on tempered distributions. Let $T\in \mathscr S'(\mathbb R^n)$ and define the Fourier transform $\hat T$ by the bracket of duality $$ \langle …

I will more comfortable with the notation $v_\epsilon=\hat{u_\epsilon}$; you have then $$ v_\epsilon(t,x)=\alpha(t,x)+\int_0^t\int e^{2πix(\xi+\epsilon\phi(s,\xi))} \hat{v_\epsilon}(s,\xi) d\xi ds=\al …

Too long for a comment. I find your question interesting for a reason linked to the proof of the Faà de Bruno formula. Let $f,g$ be two functions from $\mathbb R$ into itself. Let me assume that $f(0) …

Let $\gamma$ be defined on $\mathbb R^n$ by $\gamma (x)=e^{-π x^2}$. With $\mathcal F$ standing for the Fourier transformation defined on the Schwartz space by $$ (\mathcal F u)(\xi)=\int e^{-2iπ x\cd …

The Wiener algebra $W_n$ is the image by the Fourier transform of $L^1(\mathbb R^n)$. What is the (complex) interpolation space between $W_n$ and $L^2(\mathbb R^n)$? It is probably not true that for $ …