- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:09
Journal of the European Mathematical Society
J. Eur. Math. Soc.
JEMS
1435-9855
1435-9863
General
10.4171/JEMS
http://www.ems-ph.org/doi/10.4171/JEMS
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
15
2013
3
Limiting Sobolev inequalities for vector fields and canceling linear differential operators
Jean
Van Schaftingen
Université Catholique de Louvain, LOUVAIN-LA-NEUVE, BELGIUM
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
The estimate \[ \|{D^{k-1}u}\|_{L^{n/(n-1)}} \le \|{A(D)u}\|_{L^1} \] 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 \[ \bigcap_{\xi \in \mathbb R^n \setminus \{0\}} A(\xi)[V]=\{0\}. \] 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.
Functional analysis
Real functions
Fourier analysis
General
877
921
10.4171/JEMS/380
http://www.ems-ph.org/doi/10.4171/JEMS/380