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Let me first recall the Fourier characterization of the Lipschitz functions: Take $S$ be a compactly supported function in $\mathbb R^n$, equal to 1 near 0. We note $S_\nu$ the Fourier multiplier $S(2 …

The mother of all Gagliardo-Nirenberg inequalities is $$ \Vert u\Vert_{L^{\frac{n}{n-1}}(\mathbb R^n)}\le c_n \Vert \nabla u\Vert_{L^{1}(\mathbb R^n)}, \tag {GN} $$ where $c_n$ depends only on $n$ and …

Let $u$ be a distribution on some open subset $\Omega$ of $\mathbb R^n$. A point $(x_0,\xi_0)\in \Omega\times\mathbb S^{n-1}$ does not belong to $WF u$ when there exists a neighborhhod $V$ of $x_0$ an …

No: take a radial function in $H^s$ and not in $H^{s+\epsilon}$ for any $\epsilon >0$.

The Gevrey space is defined by a regularity condition, which can be summarized by a strong decay of the Fourier transform. A fonction $u$ belongs to $G^s$ if its Fourier transform satisfies for some $ …

Let $\kappa:\mathbb R\rightarrow\mathbb R$ be a diffeomorphism with $\kappa(0)=0$. Mimicking the change of variable formula, we would like to have $$ \int \delta(\kappa(x))\vert \kappa'(x)\vert \phi(\ …

Choosing properly the Gaussian (e.g. $e^{-π\vert x\vert^2}$) and the normalization for the Fourier transform, your equation becomes $$ \hat f(\xi)=e^{-π\vert \xi\vert^2}\hat \mu(\xi). $$ This implies …

You are talking about $B^{0,\infty}_\infty$. Take a function $u$ in the Zygmund class $B^{1,\infty}_\infty$, which the vector space of $L^\infty$ functions such that $$\exists C,\forall x,h,\quad \v …

The function $\vert x\vert^{\alpha-n}$ is radial homogeneous of degree $\alpha-n$, so its Fourier transform is radial homogeneous of degree $-(\alpha-n)-n=-\alpha$ (both locally integrable since $\alp …

For your constant coefficient operator $L(D)$, you want a fundamental solution $G$ such that $\hat G$ is a multiplier of $\mathcal S'$. This is not even true for the Laplace equation: the fundamental …

Changing variables, we have $$(pv(K)\ast f)(x)=\lim_{\delta\searrow 0} \int_{\vert x-y\vert>\delta}K(x-y) f(y) dy=\lim_{\delta\searrow 0} \int_{\vert x-G(w)\vert>\delta}K(x-G(w)) f(G(w))\vert \nabla …

Take a pseudodifferential operator with a symbol $p\in S^1_{1,0}$ and $f$ a Lipschitz-continuous function. Then the commutator $$ [p(x,D),f] $$ is bounded on $L^2$. When $f$ is $C^\infty$ with bounded …

Let $f:\mathbb C\rightarrow\mathbb C$ be a $C^\infty$ function such that $f(0)=0$. Let $1\le n\in \mathbb N$ and $s>n/2$. Then $$\forall u\in H^s(\mathbb R^n),\quad f(u)\in H^s(\mathbb R^n). $$ In pa …

In the first place your presentation must be clarified. An homogeneous distribution $u$ of degree $a$ on $\mathbb R^n$ is characterized by $$\forall \lambda >0,\quad u(\lambda x)=\lambda^au(x),\qquad …

Let me assume that $ f(t,x)=\sum_{k\ge 0} f_k(x) t^k, \quad \vert x\vert \le 1, \quad \vert t\vert < 1, $ with $f_k$ Lipschitz-continuous with an $L^\infty$ norm on $\vert x\vert \le 1$ bounded abov …