Limiting behaviour of intrinsic seminorms in fractional order Sobolev spaces
Volume 214 / 2013
Studia Mathematica 214 (2013), 101-120
MSC: Primary 46E35; Secondary 46E30, 46F12.
DOI: 10.4064/sm214-2-1
Abstract
We collect and extend results on the limit of $\sigma^{1-k}(1-\sigma)^{k}|v|_{l+\sigma,p,\varOmega}^p$ as $\sigma\to0^{+}$ or $\sigma\to1^{-}$, where $\varOmega$ is $\mathbb{R}^n$ or a smooth bounded domain, $k\in\{0,1\}$, $l\in\mathbb{N}$, $p\in[1,\infty)$, and $|\,\cdot\,|_{l+\sigma,p,\varOmega}$ is the intrinsic seminorm of order $l+\sigma$ in the Sobolev space $W^{l+\sigma,p}(\varOmega)$. In general, the above limit is equal to $c[v]^p$, where $c$ and $[\,\cdot\,]$ are, respectively, a~constant and a seminorm that we explicitly provide. The particular case $p=2$ for $\varOmega=\mathbb{R}^n$ is also examined and the results are then proved by using the Fourier transform.
