I'm attempting to understand the statement in Example 2.1 of the paper https://openreview.net/pdf?id=tFvr-kYWs_Y. In Example, denote the Sobolev space with order $s$ as $H^{s}$. Then the authors state that, by the method of real interpolation the interpolation space of $H^{s}$, i.e. $[H^{s}]^{\alpha}$, satisfies $[H^{s}]^{\alpha} \cong H^{\alpha s}$.
I have the following questions about the statement.
The author didn't define the notation $\cong$ throughout the paper, my understanding is that it means equivalent class. Is my understanding correct?
How does the method of real interpolation can lead to such a result, i.e. what is the procedure for showing it is true?
Thanks.
Typically, in this setting, $\cong$ means that both spaces are isomorphic. Typically, it means that elements in both spaces are in a one-to-one correspondence, which is linear, continuous, and with continuous reciprocal.
Interpolation is always interpolation between two spaces and with a parameter between $0$ and $1$. Here, their notation $[H^s]^\alpha$ (with $\alpha > 0$ possibly large) refers to their definition (13). They give a reference for the result you are inquiring about, so you should probably consult it.
Nevertheless, taking a step back, you can look at the space say of real-valued functions on $[0,2\pi]$. You probably know about the Sobolev spaces $H^k(0,2\pi)$ for $k \in \mathbb{N}$, which correspond to the fact that $f, f', ... f^{(k)} \in L^2(0,2\pi)$. With appropriate boundary conditions, you probably also know that, representing $f$ as the sum of its Fourier series, say $$ f(x) = \sum_n f_n \sin (n x) $$ then the norms are equivalent to $$ \| f \|_{H^k}^2 \approx \sum_n n^{2k} f_n^2. $$ Now the fractional Sobolev spaces $H^s(0,2\pi)$ for $s \in \mathbb{R}_+$ can indeed be defined by "interpolation" between the integer-order ones and satisfy $$ \| f \|_{H^s}^2 \approx \sum_n n^{2s} f_n^2. $$