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We use the notation $\hat u(\xi)=\int u(x) e^{-2iπ x\cdot \xi} dx$, meaningful for $u\in$ Schwartz space and extendable (weakly) to the huge space of tempered distributions $\mathscr S'(\mathbb R^n)$. …

One of the results due to Georges Glaeser is the following: there exists a non-negative $C^\infty$ function $f$ on the real line, flat at its zeroes, such that $\sqrt{f}$ is not $C^2$. On the other ha …

The standard Zygmund class (with $\epsilon=0$) is the Besov space $ B^1_{\infty,\infty}, $ that is using a Littlewood-Paley decomposition $$ 1=\sum_{\nu\ge 0}\varphi_\nu(\xi),\quad \varphi_\nu(\xi) =\ …

For $\phi\in C^\infty_c(\mathbb R^m)$ and duality products, we have $$ \langle\frac{\partial f}{\partial x}(x,y),\phi(x)\rangle_{\mathscr D'(\mathbb R^m),\mathscr D(\mathbb R^m)}=- \langle f(x,y),\phi …

You have an explicit formula: the symbol of $\chi_jm(D)\chi_k$ is $$ a_{jk}(x,\xi)=\chi_j(x)\iint e^{-2\pi iy\eta}m(\xi+\eta)\chi_k(y+x) dyd\eta.\tag C $$ It is difficult to handle this with $\chi_j$ …

$f$ is a continuous function thus can be considered as a distribution. For $\phi\in C^1_c(0,1)$, we have $$ \langle f',\phi\rangle=-\int_{\mathbb R} f(x) \phi'(x) dx=\lim_{\epsilon \rightarrow 0} \int …

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 …

Your formula is based on the identity for $s\in\mathbb N$, $$ \partial^s=(1+(-1)+\partial)^s=\sum_{m\ge 0}C_s^m((-1)+\partial)^m=\sum_{m\ge 0\atop 0\le k\le m}C_s^mC_m^k(-1)^{m-k}\partial^k. \tag{$\fl …

The function $\widetilde W$ is a smooth function on the open set $\Phi(B)$ and is supported in the compact set$\Phi(I_1\times I_2)$. As a result it can be extended by 0 as you wish and this is simply …

Let us start with the so-called Gagliardo-Nirenberg Inequality in $n$ dimensions, $$ \Vert u\Vert_{L^{n/(n-1)}(\mathbb R^n)}\le c_n\Vert \nabla u\Vert_{L^{1}(\mathbb R^n)}, \tag{GN}$$ an inequality th …

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 …

More a comment than an answer, but too long for a comment. First a comment on Michael Renardy's remark: there is no homogeneous function in $L^1(\mathbb R^N)$ so the first assumption is not satisfied. …

Any distribution $T$ on the real line has an anti-derivative, i.e. there exists a distribution $S$ such that $$S'=T\tag{$\ast$}.$$ Here is a constructive proof: with a given $T$, define the distribut …

Let me quote the simplest and most classical result for a global inverse function theorem, due to Hadamard and Plastock (see L. Nirenberg, Topics in Nonlinear functional analysis, Courant LN,6, 2001). …

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, …