While reading some notes due to Markus Faustmann, I came across the following exercise regarding the fractional Dirichlet problem on the unit ball.
\begin{equation*} \left\{ \begin{aligned} (-\Delta)^\alpha u &= f, && \text{on } B_1(0), \\ u &= 0, && \text{on } \mathbb{R}^N \setminus B_1(0), \end{aligned} \right. \end{equation*} where the fractional Laplacian $(-\Delta)^\alpha$ for $\alpha \in (0,1)$ is defined by \begin{equation*} (-\Delta)^\alpha u(x) := C(N,\alpha) \, \text{P.V.} \int_{\mathbb{R}^N} \frac{u(x) - u(y)}{\vert x-y \vert^{N+2\alpha}} \, dy, \end{equation*} and $f \equiv 2^{2\alpha} \Gamma(1 + \alpha)^2$.
Exercise 5.3 of the linked notes indicates we have the following solution to the above problem \begin{equation*} u(x) = (1 - \vert x \vert^2)_+^\alpha, \qquad g_+:= \max\{g,0\}, \end{equation*} and it has the following Sobolev regularity \begin{equation} u \in H^{\alpha + \frac{1}{2} - \varepsilon}(B_1(0)), \qquad\forall \, \varepsilon > 0, \end{equation} but \begin{equation} u \notin H^{\alpha + \frac{1}{2}}(B_1(0)). \end{equation}
I'm trying to show the above two inclusions. It is clear that $u \in L^2(B_1(0))$ since $u$ and the domain are bounded. What remains is to show the Sobolev-Slobodeckij seminorm $\vert u\vert_{H^\theta(B_1(0))}$ is finite for $\theta \in (-\infty, \alpha + \frac{1}{2})$, where \begin{equation*} \vert u \vert_{H^{\theta}(B_1(0))}^2 := \int_{B_1(0)} \int_{B_1(0)} \frac{(u(x) - u(y))^2}{\vert x - y \vert^{N + 2\theta}} \,dy \, dx \end{equation*}
The result is clear when $N + 2\theta \le 0$, which corresponds to $\theta \in (-\infty, -\frac{N}{2}]$. Also, using $u \in C^{0,\alpha}(\overline{B_1(0)})$, we can extend to the case $-N - 2(\theta - \alpha) > -1$, i.e., $\theta \in (-\infty, -\frac{N}{2} + \alpha + \frac{1}{2})$. Unfortunately, neither of these observations are particularly useful, since the weak formulation of the problem guarantees that $u \in \tilde{H}^\alpha(B_1(0))$.
Can somebody provide some guidance on how to proceed with showing $u \in H^{\alpha + \frac{1}{2} - \varepsilon}(B_1(0)) \setminus H^{\alpha + \frac{1}{2}}(B_1(0))$ for $\varepsilon > 0$? Thank you!