Rough path metrics on a Besov--Nikolskii type scale

It is known, since the seminal work [T. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998)], that the solution map associated to a controlled differential equation is locally Lipschitz continuous in $q$-variation resp. $1\mathrm{/}q$-H"{o}lder type metrics on the space of rough paths, for any regularity $1\mathrm{/}q\in (0,1]$. We extend this to a new class of Besov-Nikolskii-type metrics, with arbitrary regularity $1\mathrm{/}q\in (0,1]$ and integrability $p\in [q,\mathrm{\infty }]$, where the case $p\in \{q,\mathrm{\infty }\}$ corresponds to the known cases. Interestingly, the result is obtained as consequence of known $q$-variation rough path estimates.