论文标题
Vicsek Fractal上的Sobolev空间和庞加莱的不平等现象
Sobolev spaces and Poincaré inequalities on the Vicsek fractal
论文作者
论文摘要
在本文中，我们证明了Sobolev空间的几种自然方法在Vicsek分形上重合。更确切地说，我们表明，Korevaar-Schoen的度量方法，离散$ p $ edergies的限制的方法以及有线系统上Sobolev空间的限制的方法，所有方法都产生了相同的功能空间，其功能空间具有相同的标准$ p> 1 $。结果，我们证明Sobolev空间形成了真实的插值量表。我们还获得$ l^p $-poincaré的不平等，所有$ p \ ge 1 $的不平等现象。
In this paper we prove that several natural approaches to Sobolev spaces coincide on the Vicsek fractal. More precisely, we show that the metric approach of Korevaar-Schoen, the approach by limit of discrete $p$-energies and the approach by limit of Sobolev spaces on cable systems all yield the same functional space with equivalent norms for $p>1$. As a consequence we prove that the Sobolev spaces form a real interpolation scale. We also obtain $L^p$-Poincaré inequalities for all values of $p \ge 1$.
