for the function $f[u_{BS}]$.

\[lemmafuBS\] Suppose that $\nu$ is an admissible activity Lévy measure $\nu$ with the shape parameters $\alpha, D^\pm$, and $\mu$ where $\alpha<3$ and either $\mu>0, D^\pm\in\mathbb{R}$, or $\mu=0, D^- +1<0<D^+$. Suppose that $\frac12 \le \gamma<1$ and $\frac{\alpha-1}{2}<\gamma < \frac{p+1}{2p}\le1$. Then there exists a constant $C_0>0$ depending on the parameters $p,\sigma,r,T,K$ only, and such that the function $f[u_{BS}(\tau, \cdot)]$ satisfies the following estimates: $$\begin{aligned}
&&\Vert f[u_{BS}(\tau, \cdot)] \Vert_{L^p} 
\le C_0 \tau^{-(2\gamma-1)\left(\frac{1}{2} - \frac{1}{2p}\right)},\qquad 0<\tau\le T,
\\
&&\Vert f[\partial_\tau u_{BS}(\tau, \cdot)] \Vert_{L^p} 
\le C_0 \tau^{-\gamma-\frac{1}{2} + \frac{1}{2p}}, \qquad 0<\tau\le T,
\\
&&\Vert f[u_{BS}(\tau_1, \cdot)] - f[u_{BS}(\tau_2, \cdot)] \Vert_{L^p} 
\le C_0 |\tau_1-\tau_2|^{-\gamma +\frac{p+1}{2p} }, \qquad 0<\tau_1,\tau_2\le T.\end{aligned}$$

Proof. First, we consider the case of a call option, i.e. $u_{BS}=u^{call}_{BS}$ with $u_{BS}(0,x)=\Phi(K e^x) = K (e^x - 1)^+$. It is important to emphasize that $f[e^x]=0$. Hence $$f[u_{BS}] = f[u_{BS} - K e^{r\tau+x}], \quad \hbox{and}\ \ 
\partial_\tau f[u_{BS}] = f[\partial_\tau(u_{BS} - K e^{r\tau+x})].$$ In what follows, we shall denote by $C_0$ any generic positive constant depending on the parameters $p,\sigma,r,T,K$ only. With regard to Lemma \[lemma-f\] we shall estimate the $X^{\gamma-1/2}$ norm of the function $v$: $$v(\tau,x)= \partial_x \left(u_{BS}(\tau,x) - K e^{r\tau+x}\right)
= K e^{r\tau+x}( N(d_1(\tau,x)) -1),
\label{vtaux}$$ where $d_1(\tau,x) = \left(x+(r+\sigma^2/2)\tau\right)/(\sigma\sqrt{\tau})$. 