empty) proper algebraic set of ${\mathbb {P}}^k$,

- $\psi \geq \log d(\cdot,\NN_\psi)^\rho$ for some $\rho >0$.

For instance, the Hölder functions are in $\UU$, as well as the unbounded function $\log {{\textsf {Jac} \,  }}f$. We will show that $\UU \subset
L^p(\mu)$ for any $1 \leq p < + \infty$ (see subsection \[cccd\]).

[B:]{} Let $\psi \in \UU$ be a $\mu$-centered observable and ${\omega}$ be a coding map provided by theorem A. Let $\chi:= \psi \circ
  {\omega}$ and $1 \leq p < + \infty$. We denote by ${\mathbb {E}}( \chi \vert
  \CC_n  )$ the conditional expectation of $\chi$ with respect to the $(n+1)$-cylinders.

1.  there exist $\hat c_p, \lambda_p > 0$ such that $\norm { \chi   - {\mathbb {E}}( \chi \vert
      \CC_n  )  }_p \leq \hat c_p \, e^{-n\lambda_p}$ for every $n \geq 0$.

2.  $R_j(\chi):= \int_\Sigma \chi \cdot
    \chi \circ s^j \, d\nu$ satisfies $\abs{R_j(\chi)} \leq 2 \norm {\chi}_2  \hat c_2 \,
      e^{-(j-1) \lambda_2}$ for every $j \geq 1$.

The proof occupies section \[estiho\], it is based on the regularity properties of ${\omega}$ (namely the points 1, 2 of theorem A) and on the fact that $\mu$ is a Monge-Ampère mass with Hölder potentials. Theorem B allows us to 