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I am looking for references on the topic of Sobolev spaces based on $L^p$ with $0<p<1$. For instance, a natural question could be: let $u$ be a (compactly supported) distribution on $\mathbb R^n$ suc …

Let $s_1,s_2\in \mathbb R$ such that $-\frac12<s_1\le s_2$. There exists $C>0$ such that for all smooth functions $w$ , for all $r>0$, $$\operatorname{supp} w \subset(-r,r)\Longrightarrow \Vert{w}\Ve …

It is well-known that $H^{\frac d2}(\mathbb R^d)=W^{\frac d2, 2}(\mathbb R^d)$ is not included in $L^\infty(\mathbb R^d)$, but it seems that there are some logarithmic substitutes. Is it true for inst …

The standard Gagliardo-Nirenberg Inequality is $$ \Vert u\Vert_{L^{\frac{n}{n-1}}(\mathbb R^n)}\le C_n \Vert \nabla u\Vert_{L^{1}(\mathbb R^n)}, \tag{$\ast$}$$ and constitutes a key step to proving So …

It is well known that there exists a constant $C$ such that $$\forall f\in C^\infty_c(\mathbb R^3), \quad \Vert f\Vert_{L^6(\mathbb R^3)}\le C\Vert \nabla f\Vert_{L^2(\mathbb R^3)}. \tag{$\ast$}$$ Now …

It follows from the Gagliardo-Nirenberg inequality that for a locally integrable function $f$ defined on $\mathbb R^d$ (we assume $d\ge 3$) such that $\nabla f$ belongs to $L^2(\mathbb R^d)$ and $$ \ …

The standard functional isoperimetric inequality is for an integer $n\ge 1$, $$ \Vert u\Vert_{L^{\frac{n}{n-1}}(\mathbb R^n)}\le c(n)\Vert \nabla u\Vert_{L^1(\mathbb R^n)}, \quad c(n)=\frac{(\vert\mat …

Let me start with a couple of notational reminders. For $\xi\in \mathbb R^n$, $$ 1=\varphi_{0}(\xi)+\sum_{\nu \ge 1}\varphi_{\nu}(\xi),\quad \varphi_{0}\in C^\infty_c(\mathbb R^{n}),\quad \varphi_{\nu …