# Limiting Sobolev inequalities for vector fields and canceling linear differential operators

Journal of the European Mathematical Society (2013)

- Volume: 015, Issue: 3, page 877-921
- ISSN: 1435-9855

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topVan Schaftingen, Jean. "Limiting Sobolev inequalities for vector fields and canceling linear differential operators." Journal of the European Mathematical Society 015.3 (2013): 877-921. <http://eudml.org/doc/277266>.

@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 -

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