Given an operator $\mathcal{T}$ that maps from a function $f: \mathbb{R}^d\rightarrow \mathbb{R}$ to another function $f': \mathbb{R}^d\rightarrow \mathbb{R}$, we are interested in the fixed point $f^*$ (assuming that $T$ is strictly contractive so that the fixed point exists and is unique): $$ f^* = \mathcal{T}f^*. $$ One motivating example is Q-iteration in Markov decision process, where $$ f(s, a) \leftarrow \mathcal{T} f(s,a): =r(s) + \gamma \mathbf{E}_{s'\sim p(\cdot | s, a)} \left\{\max_{a'\in \mathcal{A}}f(s', a')\right\}\quad (\gamma<1). $$ Here $s\in \mathbb{R}^d$ and $a \in \mathcal{A}$ denote the state and action, where $\mathcal{A}$ is a discrete set. Meanwhile, $r(s): \mathbb{R}^d \rightarrow \mathbb{R}$ is the reward function, $p(\cdot | s, a)$ is the transition kernel, that is, a conditional distribution of the next state $s'$ given the current state and action, and $a'$ is the action taken at the next state $a'$.

Without the max operator, the above fixed-point iteration corresponds to a linear operator. In such a simplified case, what can we say about the properties of the fixed point $f^*$, assuming properties of the operator, which depends on the transition kernel and reward function? (For example, the smoothness of $f^*$, or more generally, its Sobolev- or Hölder-norm.)

What about the case with the max operator? More generally, what about general strictly contractive nonlinear operators?