For $1\leq p<\infty$ an approach to define fractional Sobolev spaces is by Sobolev-Slobodeckij spaces a generalisation of Hölder continuity. For example letting $U\subset\mathbb{R}^n$ then,

$ \left\|u\right\|^p_{W^\mu_p(U)} = \left\|u\right\|^p_{W^{\lfloor\mu\rfloor}_p(U)} + \sum \int_U \int_U \frac{|D^\alpha u(x)-D^\alpha u(y)|^p}{|x-y|^{n+p[\mu]}}dxdy $

where $[\mu]=\mu-\lfloor\mu\rfloor$ and the sum is taken over all multi-indices $\alpha$ with $|\alpha|=\lfloor\mu\rfloor$

This is from Chapter 14 of The mathematical theory of finite element methods By Susanne C. Brenner, L. Ridgway Scott.

Does the above hold for $p=\infty$? For example, for $p=\infty$ do we have (or something similar),

$ \left\|u\right\|_{W^\mu_p(U)} = \left\|u\right\|_{W^{\lfloor\mu\rfloor}_p(U)} + \sup \sup_U \sup_U \frac{|D^\alpha u(x)-D^\alpha u(y)|}{|x-y|^{[\mu]}} $

where the $\sup$ is taken over all multi-indices $\alpha$ with $|\alpha|=\lfloor\mu\rfloor$. Can this be shown by considering the limit of the case $p<\infty$ as $p\rightarrow\infty$?

Thanks in advance.


Yes, it is Hölder spaces and can be regarded as SS spaces for $p=\infty$. Actually, for natural $\mu$ more correct for functions analysis are Zygmund spaces (with differences of th second order in the definition). They are special cases of Besov spaces, which are defined for $0 < p\le \infty$, $\mu\in \mathbb R$. They coincide with corresponding Sobolev-Slobodetskij in some cases. But there are some differences between cases $p<\infty$ and $p=\infty$. For example, Holder spaces are not separable.

On the second question, shown what? It is a definition for $p=\infty$.

  • $\begingroup$ Can you give a reference for your claims? $\endgroup$ – shuhalo Jun 23 '18 at 15:50

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