Sharp trace Hardy–Sobolev inequalities and fractional Hardy–Sobolev inequalities
Sharp trace Hardy–Sobolev inequalities and fractional Hardy–Sobolev inequalities
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DOI：
10.1016/j.jfa.2015.11.016
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发表时间：
2016-06
影响因子：
1.7
通讯作者：
K. Tzirakis
中科院分区：
文献类型：
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作者：
K. Tzirakis
在这项工作中，我们建立了尖锐的加权迹 Hardy 不等式，其迹剩余项涉及由奇异对数权重校正的临界 Sobolev 指数。我们证明这个权重是最优的，因为对于更奇异的权重，不等式不成立。然后，我们应用这些结果来推导尖锐的 Hardy 不等式以及与拉普拉斯分数 s 次方 s ε (0, 1) 相关的相对改进。特别是，我们处理在有界域上定义的这种类型的两个不同运算符。由此可见，与两个不同分数拉普拉斯相关的哈代不等式共享相同的最佳常数，并且它们都可以通过添加涉及相同最佳对数校正的索博列夫型余数项来锐化。还考虑 Hardy 型余数项。我们的结果与早期的非分数对应结果直接一致，其中 s= 1，即标准拉普拉斯算子。
In this work we establish sharp weighted trace Hardy inequalities with trace remainder terms involving the critical Sobolev exponent corrected by a singular logarithmic weight. We show that this weight is optimal in the sense that the inequality fails for more singular weights. Then we apply these results to derive sharp Hardy inequalities and relative improvements associated with fractional s-th powers of the Laplacian, s∈(0, 1). In particular, we deal with two different operators of this type, defined on bounded domains. It follows that Hardy inequalities associated with two different fractional Laplacians share the same best constant as well as they can be both sharpened by adding Sobolev type remainder term involving the same optimal logarithmic correction. Hardy type remainder terms are also considered. Our results are in direct accordance with earlier results for their non-fractional counterpart where s= 1, that is the standard Laplacian.