A MILD ROUGH GRONWALL LEMMA WITH APPLICATIONS TO NON-AUTONOMOUS EVOLUTION EQUATIONS

The paper’s Lemma 4.2 states exactly the claimed bound for (z,G(·,z·)) ∈ D^γ_{X,α} with constants C1, C2 and parameters Φ1, Φ2, Φ3, ν, κ, and uses the same discrete one-step estimate and iteration scheme. The model’s solution reproduces the same structure: (i) identifies z′=G(·,z·) as the Gubinelli derivative (as in the paper’s setup), (ii) derives local bounds on short intervals via the Kato–Tanabe smoothing estimates (2.3) and rough convolution estimates, (iii) collects them into the inequality of the form (4.3), and (iv) iterates along a partition to obtain the stated exponential bound with the same constants. The paper’s formulas for Φ1, Φ2, Φ3 and C1, C2 match those in the model solution, and the iteration criterion 2CΦ2 > 1 − Cκ^ν Φ3 > 0 is the same. The only minor quibble is a notational slip in the model’s description of how U_{v,u} contributes a factor |v−u|^{-σ} (the model briefly wrote −(γ+σ)), but this does not change the iteration or the final bound. Overall, both are correct and essentially the same proof, differing only in presentation details. See Lemma 4.2 and its proof, including the local inequality (4.3), the constants, and the Kato–Tanabe estimates (2.3).