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/q-H¨older-type metrics on the space of rough
paths, for any regularity 1/q ∈ (0, 1].
We extend this to a new class of Besov–Nikolskii-type metrics, with arbitrary regularity 1/q ∈ (0, 1] and integrability p ∈ [q, ∞], where the case
p ∈ {q,∞} corresponds to the known cases. Interestingly, the result is obtained as a consequence of known q-variation rough path estimates