Limiting Sobolev inequalities for vector fields and canceling linear differential operators

@article{VanSchaftingen2013,
abstract = {The estimate $\left\Vert D^\{k-1\}u\right\Vert _\{L^\{n/(n-1)\}\}\le \left\Vert A(D)u\right\Vert _\{L^1\}$ is shown to hold if and only if $A(D)$ is elliptic and canceling. Here $A(D)$ is a homogeneous linear differential operator $A(D)$ of order $k$ on $\mathbb \{R\}^n$ from a vector space $V$ to a vector space $E$. The operator $A(D)$ is defined to be canceling if $\bigcap _\{\xi \in \{\mathbb \{R\}\}^n\setminus \lbrace 0\rbrace \} A(\xi ) [V] = \lbrace 0\rbrace $. This result implies in particular the classical Gagliardo–Nirenberg–Sobolev inequality, the Korn–Sobolev inequality and Hodge–Sobolev estimates for differential forms due to J. Bourgain and H. Brezis. In the proof, the class of cocanceling homogeneous linear differential operator $L(D)$ of order $k$ on $\mathbb \{R\}^n$ from a vector space $E$ to a vector space $F$ is introduced. It is proved that $L(D)$ is cocanceling if and only if for every $f\in L^1(\mathbb \{R\}^n;E)$ such that $L(D)f=0$, one has $f\in \dot\{W\}^\{-1,n/(n-1)\}(\mathbb \{R\}^n;E)$. The results extend to fractional and Lorentz spaces and can be strengthened using some tools of J. Bourgain and H. Brezis.},
author = {Van Schaftingen, Jean},
journal = {Journal of the European Mathematical Society},
keywords = {Sobolev embedding; overdetermined elliptic operator; compatibility conditions; homogeneous differential operator; canceling operator; cocanceling operator; exterior derivative; symmetric derivative; homogeneous Triebel−Lizorkin space; homogeneous Besov space; Lorentz space; homogeneous fractional Sobolev-Slobodeckiĭ space; Korn-Sobolev inequality; Hodge inequality; Saint-Venant compatibility conditions; Sobolev embedding; overdetermined elliptic operator; compatibility conditions; homogeneous differential operator; canceling operator; cocanceling operator; exterior derivative; symmetric derivative; homogeneous Triebel-Lizorkin space; homogeneous Besov space; Lorentz space; homogeneous fractional Sobolev-Slobodeckiĭ space; Korn-Sobolev inequality; Hodge inequality; Saint-Venant compatibility conditions},
language = {eng},
number = {3},
pages = {877-921},
publisher = {European Mathematical Society Publishing House},
title = {Limiting Sobolev inequalities for vector fields and canceling linear differential operators},
url = {http://eudml.org/doc/277266},
volume = {015},
year = {2013},
}

TY - JOUR
AU - Van Schaftingen, Jean
TI - Limiting Sobolev inequalities for vector fields and canceling linear differential operators
JO - Journal of the European Mathematical Society
PY - 2013
PB - European Mathematical Society Publishing House
VL - 015
IS - 3
SP - 877
EP - 921
AB - The estimate $\left\Vert D^{k-1}u\right\Vert _{L^{n/(n-1)}}\le \left\Vert A(D)u\right\Vert _{L^1}$ is shown to hold if and only if $A(D)$ is elliptic and canceling. Here $A(D)$ is a homogeneous linear differential operator $A(D)$ of order $k$ on $\mathbb {R}^n$ from a vector space $V$ to a vector space $E$. The operator $A(D)$ is defined to be canceling if $\bigcap _{\xi \in {\mathbb {R}}^n\setminus \lbrace 0\rbrace } A(\xi ) [V] = \lbrace 0\rbrace $. This result implies in particular the classical Gagliardo–Nirenberg–Sobolev inequality, the Korn–Sobolev inequality and Hodge–Sobolev estimates for differential forms due to J. Bourgain and H. Brezis. In the proof, the class of cocanceling homogeneous linear differential operator $L(D)$ of order $k$ on $\mathbb {R}^n$ from a vector space $E$ to a vector space $F$ is introduced. It is proved that $L(D)$ is cocanceling if and only if for every $f\in L^1(\mathbb {R}^n;E)$ such that $L(D)f=0$, one has $f\in \dot{W}^{-1,n/(n-1)}(\mathbb {R}^n;E)$. The results extend to fractional and Lorentz spaces and can be strengthened using some tools of J. Bourgain and H. Brezis.
LA - eng
KW - Sobolev embedding; overdetermined elliptic operator; compatibility conditions; homogeneous differential operator; canceling operator; cocanceling operator; exterior derivative; symmetric derivative; homogeneous Triebel−Lizorkin space; homogeneous Besov space; Lorentz space; homogeneous fractional Sobolev-Slobodeckiĭ space; Korn-Sobolev inequality; Hodge inequality; Saint-Venant compatibility conditions; Sobolev embedding; overdetermined elliptic operator; compatibility conditions; homogeneous differential operator; canceling operator; cocanceling operator; exterior derivative; symmetric derivative; homogeneous Triebel-Lizorkin space; homogeneous Besov space; Lorentz space; homogeneous fractional Sobolev-Slobodeckiĭ space; Korn-Sobolev inequality; Hodge inequality; Saint-Venant compatibility conditions
UR - http://eudml.org/doc/277266
ER -