一类非线性半群的超压缩性、Nash不等式和从属性
Hypercontractivity, Nash inequalities, and subordination for classes of nonlinear semigroups
摘要：
给出了非线性半群$\{T_t\}$的超压缩性的一个适当的概念，并证明了它的生成元$H$的Gagliardo-Nirenberg不等式，只要能量泛函$(u,Hu)$具有次齐性.我们利用这一事实证明，对于由$p$ -Laplacian-型算子生成的半群，超压缩性蕴涵超压缩性。当Bernstein函数为$f(x)=x^α$时，我们引入了从属非线性半群的概念，并给出了相应生成元的显式表达式.它表明，在某些情况下，超收缩仍然成立的从属半群，因此，纳什型不等式成立，以及从属发电机。
英文摘要：
A suitable notion of hypercontractivity for a nonlinear semigroup $\{T_t\}$ is shown to imply Gagliardo--Nirenberg inequalities for its generator $H$, provided a subhomogeneity property holds for the energy functional $(u,Hu)$. We use this fact to prove that, for semigroups generated by operators of $p$--Laplacian--type, hypercontractivity implies ultracontractivity. We then introduce the notion of subordinated nonlinear semigroups when the corresponding Bernstein function is $f(x)=x^α$, and write down an explicit formula for the associated generator. It is shown that, in certain cases, hypercontractivity still holds for the subordinated semigroup and, hence, that Nash--type inequalities holds as well for the subordinated generator.
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