I'm studying this article https://projecteuclid.org/download/pdf_1/euclid.twjm/1500574954 and I'm having problems understanding the proof of lemma 3.
Let me recall some of the criminals involved. Let $(u_t)_{t \in [0,T]}$ be a simple bounded process of the form
$$ u = \sum_{j = 0}^{n-1} F_j 1_{(t_s, t_{j+1}]}, $$ let us define $$ u^\epsilon_t = \frac{1}{2 \epsilon}\int_{t - \epsilon}^{t+\epsilon} u_r \ dr. $$ Let us define the seminorm $$ \lvert| \phi |\rvert_K^2 = \int_0^T \phi(s)^2 K(T,s)^2 \ ds + \int_0^T \left(\int_s^T \left | \phi(t) - \phi(s) \right | (t-s)^{H - \frac{3}{2}} \ dt \right)^2 \ ds $$ where $K$ satisfies $$ \left | K(t,s) \right | \leq c((t-s)^{H - \frac{1}{2}} + s^{H - \frac{1}{2}}) $$ and $$\left | \frac{\partial K}{\partial t} (t,s) \right | \leq c (t-s)^{H - \frac{3}{2}}$$ where $H < \frac{1}{2}$. Let $\mathcal{H}_K$ be the completion of the set of step functions with respect to the norm defined above. Then at page 614 it is stated that $u_t^\epsilon$ converges to $u$ in $\mathbb{D}^{1,2}(\mathcal{H}_K)$. The key step that I'm missing is the integral estimate done at page 615, in fact I obtain and estimate that also involves the integral: $$ \int_{t_i + 2 \epsilon}^{t_{i+1} - 2 \epsilon} \left( \int_{s}^{t_i + 2 \epsilon} \left| u_t^\epsilon - u_s^\epsilon \right | (t-s)^{H - \frac{3}{2}} \ dt \right)^2 \ ds $$ and I am not able to state that this integral converges to zero. Any suggestions?