Can we interpret fractional Sobolev spaces in terms of fractional derivatives?

Question

Let $1 \leq p < \infty$, $0<s<1$, and $\Omega \subseteq R^n$ be a domain. The Banach space $W^{s,p}(\Omega)$ is defined as $$W^{s,p}(\Omega) := \left\{ f \in L^p(\Omega) \colon \int_{\Omega \times \Omega} \frac{ |f(x)-f(y)|^p }{ |x-y|^{ s p + n } } dx dy < \infty \right\}.$$

I am wondering whether this "fractional Sobolev space" can be interpreted in terms of fractional derivatives in any manner. Can we interpret functions in $W^{s,p}(\Omega)$ as having a certain fractional derivative of a certain regularity class?

Answer 1

Yes, such an interpretation exists, at least in the following case.

Take $p = 2$, $n = 1$ and $\Omega = (0,1)$. Then, for $0 < s < 1$, $$ \| f \|_{H^s(0,1)} \approx \| \partial_t^s f \|_{L^2(0,1)} $$ where $\partial_t^s f$ is the Caputo fractional derivative of $f$. This interpretation and result is proved in Section 3 of [1].

Still in 1D, there are results for $p \neq 2$, for example Theorem 18.3 in [2], which is stated using fractional primitive instead of fractional derivatives.

I do not know about the higher-dimensional case.

[1] Gorenflo, Rudolf; Luchko, Yuri; Yamamoto, Masahiro, Time-fractional diffusion equation in the fractional Sobolev spaces, Fract. Calc. Appl. Anal. 18, No. 3, 799-820 (2015). ZBL1499.35642.

[2] Samko, St. G.; Kilbas, A. A.; Marichev, O. I., Fractional integrals and derivatives: theory and applications. Transl. from the Russian, New York, NY: Gordon and Breach. xxxvi, 976 p. (1993). ZBL0818.26003.