This question arose via the helpful comments on this earlier question.

In Hairer's theory of regularity structures, fixed point problems are first solved in certain spaces $D^\gamma$ which consist of truncated Taylor-like series. A commenter on a previous post explained that the Picard iteration occurs by applying the operator in question, say $A$, and then truncating at each step to stay in $D^\gamma$.

Q1. How does one know that the reconstruction operator applied to this sequence of truncated iterates converges to a genuine fixed point of $A$ on distributions when $A$ is actually defined on distributions?

The truncation is what is tripping me up. My intuition is that once you apply $A$ and the regularity of one of your functions goes above 1, it is expressible in terms of the terms of regularity less than the $\gamma$ we've prescribed for the problem. But this can't be the case exactly because at one point Hairer takes 3/2 as the $\gamma$.

Q2. If this is indeed the correct explanation – that $A$ applied to a term of high enough regularity is expressible as a modelled distribution with coefficients in some Hölder space – how does one prove this?

Thank you!