Hardy-Sobolev inequalities for vector fields and canceling linear differential operators

By Pierre Bousquet, Jean Van Schaftingen. 2013

80 citations recorded by the source corpus.

The estimate \[ \norm{D^{k-1}u}_{L^{n/(n-1)}} \le \norm{A(D)u}_{L^1} \] 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 \(\R^n\) from a vector space \(V\) to a vector space \(E\). The operator \(A(D)\) is defined to be canceling if \[ \bigcap_{\xi \in \R^n \setminus \{0\}} A(\xi)[V]=\{0\}. \] 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 \(\R^n\) 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 \(f \in L^1(\R^n; E)\) such that \(L(D)f=0\), one has \(f \in \dot{W}^{-1, n/(n-1)}(\R^n; E)\). The results extend to fractional and Lorentz spaces and can be strengthened using some tools of J. Bourgain and H. Brezis.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/99ec258b9571e64a2ea61c4bd99b32c13c091f69

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In Applied Mathematics

  1. Franz Gmeineder
  2. Bogdan Raiță
  3. Adolfo Arroyo-Rabasa
  4. Guido De Philippis
  5. Filip Rindler
  6. Jean Van Schaftingen
  7. Enrico Pasqualetto
  8. Nicola Gigli

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