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I am trying to prove the Gagliardo--Nirenberg type inequality: $$ \Vert\nabla u\Vert_{L^{2p}(\mathbb{R}^{N})}\leq c|u|_{\operatorname*{BMO}% (\mathbb{R}^{N})}\Vert\nabla^{2}u\Vert_{L^{p}(\mathbb{R}^{N …

Given $1\leq p<\infty$, the homogeneous Sobolev space $\dot{W}^{1,p} (\mathbb{R}^{N})$ is defined as the space of all functions $u\in L_{\operatorname*{loc}}^{1}(\mathbb{R}^{N})$ such that the distrib …

I want to write a function in terms of its mollification and higher order forward differences. Given a function $u:\mathbb{R}\rightarrow\mathbb{R}$ and $h>0$, we set $u_{h}(x):=\frac{1}{h}u\left( \fr …

In his book Singular Integrals and Differentiability Properties of Functions, Stein constructs an extension operator $\mathcal{E}:W^{m,p}(\Omega)\rightarrow W^{m,p}(\mathbb{R}^{N})$, $m\in\mathbb{N}$, …

I am reading the paper "Bounded Mean Oscillation and Sobolev Spaces" by Robert s. Strichartz, Indiana University Mathematics Journal , 1980, Vol. 29, pp. 539-558. In this paper he defines the space $I …