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向量场的极限Sobolev不等式与消去线性微分算子

Limiting Sobolev inequalities for vector fields and canceling linear differential operators

Jean Van Schaftingen

备注：
40 pages, incorporated corrections suggested by the referee
发表信息：
J. Eur. Math. Soc. (JEMS) 15 (2013), no. 3, 877-921
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摘要：

证明了估计[\lVert D ^{k-1} u \rVert_{L^{n/(n-1)}} \le \lVert A（D）u \rVert_{L^1}]成立，当且仅当（A（D））是椭圆的且相消的。这里（A（D））是（\mathbb{R}^n）上从向量空间（V）到向量空间（E）的（k）阶齐次线性微分算子（A（D））。如果[\bigcap_{ξ\in \mathbb{R}^n \setminus {0}} A（D）[V]={0}，则运算符（A（D））被定义为取消。]这个结果特别包含了经典的Gagliardo-Nirenberg-Sobolev不等式、Korn-Sobolev不等式以及J.Bourgain和H.布雷齐斯在证明中，引入了从向量空间（E）到向量空间（F）的（\mathbb{R}^n）上的（k）阶余消去齐次线性微分算子类（L（D））.证明了（L（D））是余消去的当且仅当对每个（fPDMATHABMS L^1（\mathbb{R}^n; E））使得（L（D）f = 0）有（fPDMATHABMS \dot{W}^{-1, n/(n-1)}（\mathbb{R}^n; E））.所得结果推广到分数阶空间和Lorentz空间，并可利用J. Bourgain和H.布雷齐斯

英文摘要：

The estimate [\lVert D^{k-1}u\rVert_{L^{n/(n-1)}} \le \lVert A(D)u \rVert_{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_{ξ\in \mathbb{R}^n \setminus {0}} A(ξ)[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.

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