Assume a pure jump L\'evy process $X=(X_t)_{t\in [0,T]}$ with L\'evy measure $\nu$ and a Borel function $f:\rm{\bf R} \to \rm{\bf R}$ with $f(x+X_s)\in L_1$ for $(s,x)\in [0,T]\times \rm{\bf R}$. Define $F:[0,T]\times \rm{\bf R}\to \rm{\bf R}$ by
$F(t,x) := E f(x+X_{T-t})$ and the vector-valued gradient \[ D_J F:[0,T) \times \rm{\bf R} \to L_0(\rm{\bf R}\setminus \{0\} )
\quad \mbox{by} \quad
D_J F(t,x) := \left \{ z \mapsto \frac{F(t,x+z) - F(t,x)}{z} \right \} \] known from Malliavin calculus and non-local PDEs. If $\rho$ is a finite Borel measure on $\rm{\bf R}$ sharing the small ball estimate $\rho([-r,r])\le c r^\varepsilon$ for some $\varepsilon \ge 0$ and if the coupling property $\| P_{z+X_s} - P_{X_s}\|_{TV} \le d |z| s^{-\frac{1}{\beta}}$ holds for some $\beta \in (0,2]$, then in \cite{1} we prove \[ \left \| (T-t)^\alpha \sup_{x\not = 0} \left | \int_{\rm{\bf R}\setminus \{0\}} (\partial_J F(t,x))(z) d\rho(z)
\right | \right \|_{L_q((0,T],\frac{d t}{T-t})}
\le C \| f \|_{{\rm Hoel}_{\eta,q}},\] where $f$ belongs to the Besov space ${\rm Hoel}_{\eta,q}(\rm{\bf R})$ with $(\eta,q)\in (0,1-\varepsilon)\times [1,\infty]$, $X\subseteq L_{\eta+\gamma}$ for some $\gamma>0$, and for $\alpha:= \frac{1-(\varepsilon+\eta)}{\beta}>0$.
The exponent $\alpha$ is best possible. The estimate applies to stable like processes. Applications to the predictable representation property on the L\'evy-It\^o space and the path-regularity of the gradient $D_JF$ are given.


\begin{thebibliography}{9}
\bibitem{1} S.~Geiss and T.~Nguyen:
On Riemann-Liouville type operators, BMO, gradient estimates in the L\'evy-It\^o space, and approximation,
arXiv:2009.00899.\smallskip
\end{thebibliography}