| Summary: | 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.
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