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In the first place, you can write $ \mathbf 1_\Omega=\sum_{k\ge 1}\chi_k, \ \chi_k\in C^\infty_c(\Omega). $ Let $\nu\in C^\infty(\mathbb R^n)$, vanishing on $B(0,1/2)$, equal to 1 on $B(0,1)^c$: let u …

The problem is indeed coming form the fact that singular integrals, such as the Hilbert transform, although bounded on $L^p$ for $1<p<+\infty$ are failing to be bounded on $L^1$ or $L^\infty$. Howev …

If I understand your question correctly, you speak about interior regularity. Let me quote a classical result for linear elliptic equations with $C^\infty$ coefficients, even true for pseudo-different …

No. The operator $(-∆)^s$ is the Fourier multiplier $\vert \xi\vert^{2s}$ so that, say for $f$ in the Schwartz space whose Fourier transform vanishes near the origin, we have $$ \Vert(-∆)^{-s} f\Vert …

You note first that $$\text{div}(\omega_{\epsilon}\ast u_{\epsilon})= \omega_{\epsilon}\ast\text{div }u_{\epsilon}=0, $$ so that, at least formally, $$ \langle(\omega_{\epsilon}\ast u_{\epsilon})\cdot …

You can always consider the second (distribution) derivative of a continuous function and require that it is non-negative, which means that for all $T\in \mathbb R^n$ and all $\phi\in \mathscr D(\Omeg …

Let us set $L_1=\sum_{1\le j\le m}X_j^2$, where the $X_j$ are smooth real vector fields satisfying Hörmander's condition so that $L_1$ is subelliptic with an estimate $$ \Vert L_1 u\Vert_{H^s}\ge C\Ve …

About your second question. If I understand things correctly, you want to solve the Dirichlet problem $$ \Delta u = \operatorname{div} F, \quad u_{\vert \partial \Omega}=0. $$ Since your open set is s …

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 …

Writing $$ \langle-\Delta u + au, u\rangle_{L^2(\Omega)}=\langle f, u\rangle_{L^2(\Omega)}, $$ using the Dirichlet boundary condition, you get $$ \Vert \nabla u\Vert_{L^2(\Omega)}^2+\langle au, u\rang …

I guess that your question is ill-formulated: you have to respect the homogeneity on both sides of the inequality. Let us say that for $f$ in the Schwartz space you always have $$ \Vert f\Vert_{W^{t,q …

Let me start with a constant coefficient operator $$ P(D)=\sum_{1\le j, k\le n} a_{jk} D_jD_k,\quad D_j=\frac{\partial }{i\partial x_j}. $$ Note that in two dimensions, you have elliptic operators wit …

Yes you do have a generalization of your elliptic inequality to the $L^p$ case for $p\in (1,+\infty)$. In fact the operators with symbols in the class $S^0_{1,0}$ (as in your question) are bounded on …

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 …

There are two different parts in elliptic regularity theory. The first and easier is interior regularity, which can be proven for $p\in (1,+\infty)$ essentially by the same method as for $p=2$, usin …