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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 …

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 $ …

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 define the Fourier transform of a function $u$, say in the Schwartz space $\mathscr S(\mathbb R)$ as $ \hat u(\xi)=\int_{\mathbb R} e^{-2iπ x\cdot \xi} u(x) dx. $ The Hilbert transform $\maths …

The Wiener algebra $\mathcal W$ is defined as $\text{Fourier}(L^1(\mathbb R))$, i.e. the image by the Fourier transform of $L^1(\mathbb R)$. Riemann-Lebesgue's lemma ensures that $$ \mathcal W\subset …

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 …

Let $f,g$ be tempered distributions on $\mathbb R^n$ such that $g\in \mathscr O_M$, the so-called multipliers space: $g$ is a smooth function such that $$\forall \alpha, \exists N_\alpha\ge 0,\quad \ …

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 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 …

Your function is, with $P_d$ homogeneous harmonic polynomial of degree $d$ in $n$ variables, $$ u(x)=\frac{P_d(x)}{\vert x\vert^{n+d}}. \tag{1} $$ This is an homogeneous distribution of degree $-n$ o …

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 …

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 …

You need to check Condition (ii) in Definition 8.2.2 in the first volume of Hörmander's ALPDO. Let us note $f(x+i0)$ the limit-distribution of your question and let $\Gamma$ be its wave-front-set. Let …