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Too long for a comment. I would say that your problem is a semi-global solvability question. You will find in chapter 26 of Hörmander’s ALPDO a precise definition for that property which suits well th …

The answer to your question is negative. Take the smooth function $\chi$ defined by $$ \chi(t)=H(t) e^{-t^{-1}-t^2}, \quad H=\mathbf 1_{(0,+\infty)}. $$ This function is in $L^1(\mathbb R)$, $C^\infty …

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 …

No. Take for instance the following ODE where $0$ is a regular singular point, $$x y'=\lambda y.$$ The functions $cx_+^{\lambda}$ are solutions and are not analytic. If $\lambda$ is not a non-negative …

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 …

If $T$ is a distribution on $\mathbb R^n$ such that $\nabla T$ belongs to $L^1_{\text{loc}}$, the isoperimetric inequality implies that $$ T\in L^\frac{n}{n-1}_{\text{loc}}. \tag{1} $$ To prove this, …

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 …

Too long for a comment. I guess that your question is "If I know that $\lvert D_x\rvert^{-\alpha} f$ belongs to $\mathrm{BMO}$, what can I say about the regularity of $f$?" I guess that $\alpha$ belon …

By the Morse Lemma, using your assumption (3), you can find a smooth change of variables $x\mapsto y$ such that $$ f(x)=\frac12 f''(0) y^2 $$ in a neighborhood of 0. This entails the integrability con …

Let $H=\mathbf 1_{\mathbb R^+}$ be the Heaviside function and let us define $g$ by $ g(x)=H(x) x^{-1/2}. $ We note that $g$ is homogeneous with degree $-1/2$ and thus its Fourier transform is homogene …

Let $U$ be an open subset of $\mathbb R^d$ and let $u\in \mathscr D'(U)$. Then we have $$ (\text{supp } u)^c=\{x\in U, \exists V \text{open neighborhood of $x$ such that}\ u_{\vert V}=0\}, \tag{1}$$ …

Too long for a comment. For $u$ in $C^s$, $s\in (0,1)$, you can indeed define $u^2$ and then the distribution-derivative of $u^2$, which belongs to $B^{s-1}_{\infty,\infty}$. Now that does not define …

Consider $\rho$ be a $C^\infty$ function supported in $(-1/8,1/8)$ with integral 1 and set $ w=\chi_I\ast \rho, $ so that, for $n\ge 1$, we have $$ w^{(n)}(x)=\bigl(\chi_I\ast \rho^{(n)}\bigr)(x)= \bi …

The operator $(-∆)^s$ is the Fourier multiplier $\vert\xi\vert^{2s}$ and thus deserves to be denoted by $\vert D\vert^{2s}$. Now the distribution $u_\alpha(x)=x_+^\alpha$ is homogeneous with degree $\ …