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The answer by Bazin (https://mathoverflow.net/users/21907/bazin), Faa di Bruno's formula for vector valued functions, URL (version: 2012-09-04): https://mathoverflow.net/q/106339 is providing a formul …

Your functions $u_j$ are solutions of a semi-linear elliptic equation and, for $p>2$ the function $t\mapsto\vert t\vert^{p-1}t=\phi(t)$ is $C^2$; as a consequence, each $u_j$ is $C^\alpha$ with some p …

Too long for an additional comment. I guess that you can keep the assumption $\hat P, \hat Q$ Hermitian and require $$ [\hat P, \hat Q]=1/(2πi), $$ as it is the case with the prototypical example $ \h …

Let us assume that $p=2$, and let us consider $$ \cup_{s>0} H^s(\mathbb R^d)\subset H^0(\mathbb R^d)=L^2(\mathbb R^d). $$ The above inclusion is strict. Let us consider $u\in L^2(\mathbb R^d)$ define …

If I understand things correctly, $u$ is vanishing on some non-empty open subset. Moreover, you have pointwisely on $\mathbb R^3$ the differential inequality $$ \lvert \Delta u\rvert\le C\lvert u\rve …

There exist $c>0$ and $s>0$, such that for all smooth functions $v$ compactly supported in $\Omega$, $$ \sum_{1\le j\le m}\Vert X_jv\Vert_{L^2}\ge c\Vert v\Vert_{W^{s,2}}. $$ The largest (i.e. the bes …

I believe that for $n=3$ the optimal Hardy inequality is $$ \int_{\mathbb R^3}\vert(\nabla w)(y)\vert^2 dy\ge \frac14 \int_{\mathbb R^3}\frac{\vert w(y)\vert^2}{\vert y\vert^2} dy, $$ say for $w$ in t …

Let $\mathbb H$ be a Hilbert space and let $A\in \mathcal B(\mathbb H)$ such that the spectrum of $A$ does not meet a closed half-line issued from 0 in the complex plane. Then I guess that $ A=\exp L …

A most important result is missing in the previous answers, namely the characterization by Lars Hörmander of hypoellipticity in his seminal paper, On the theory of general partial differential operato …

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 …

Although I agree with the above answer, I would like to point out the important scaling properties linked to $BV$ functions. Let us consider a function $u$ in $L^1_{\text{loc}}(\mathbb R^n)$ such that …

I want first to change your notations, sticking to the usual variables $x,\xi$ in the phase space. As a general statement about pseudo-differential operator with a symbol $a(x,\xi)$, I wish to write $ …

Let $u\in L^2(\mathbb R^n)$: then $u\ast u$ is a bounded continuous function. Let me assume now that $u\ast u$ is compactly supported. Is there anything relevant that could be said on the support of $ …

A few reminders: [1] The dual space of $\mathscr D(\mathbb R^n)= C^\infty_c(\mathbb R^n)$ ($C^\infty$ functions with compact support) is $\mathscr D'(\mathbb R^n)$ (distributions on $\mathbb R^n$). [2 …

Denis Serre's answer is just perfect. Let me add a couple of examples of distributions that can be squared: With $H$ the Heaviside function, define $\operatorname{Log}(x+i0)=\ln(\vert x\vert)+i\pi H( …