@GiorgioMetafune It seems like you're right. I feel quite stupid now for noticing that earlier.

The first inequality has a $r$, not $R$ in the last term. Nicolaescu is basically claiming that the $(j, p)$-norm is controlled by $\epsilon$ times the $(m, p)$ norm plus $\epsilon^{...}$ times the norm on any compact subdomain. The result from Adams-Fournier only says some compact subdomain which might even depend on $\epsilon$.

@Hannes Oh that's a mistake, thanks for pointing it out. I fixed it.

I've moved this question from math.stackexchange (and deleted the question there).

@LaurentMoret-Bailly And the way to see that it is not epi in the category of schemes is to consider $f, g: \mathbb{A}^1 \to \operatorname{Spec} k[\epsilon]/\epsilon^2$ sending $t \mapsto \epsilon$ and $t \mapsto 0$.

@LaurentMoret-Bailly are you saying that the normalization of the cusp is not a monomorphism (in the category of reduced affine $\mathbb{C}$-schemes)? It seems like monomorphism and injectivity should be equivalent because morphisms of varieties are determined by their map on topological spaces (so injective => monomorphism) and monomorphism have to be injective (otherwise you have a contradiction by using two different maps from $\operatorname{Spec} \mathbb{C}$).

What makes you think that the curve should lie on one side of the circle? I think generically the curve will cross the circle. If the curvature of your curve $\gamma$ is increasing at time $t = 0$ then $\gamma(t)$ should lie inside the circle for small $t > 0$ and outside for "small" $t < 0$ and vice versa if the curvature is decreasing at 0.

@ZachTeitler That was part of the question... though it seems to me that the statement and proof are very similar if not the same as the lemma that you referenced (and I haven't had time yet to look at in detail)