- $\psi \geq \log d(\cdot,\NN_\psi)^\rho$ on ${\mathbb {P}}^k$ for some $\rho >0$.

The Hölder functions belong to $\UU$. Examples of unbounded observables are:

1.  the functions $\psi = \log \abs Q - q \log \norm{\cdot}$, where $Q$ is a $q$-homogeneous polynomial on ${\mathbb {C}}^{k+1}$. Here the algebraic subset $\NN_\psi$ is the zero set of $Q$.

2.  the functions $\psi = \log \norm{\Lambda^j d_x f}$ ($1
      \leq j \leq k$), where $\Lambda^j d_x f$ is the $j$-exterior power of the differential $d_x
      f$. In particular, $\log {{\textsf {Jac} \,  }}f \in \UU$ (take $j = k$).

The conditions of definition \[unb\] are easy to verify for these functions, the last one is a consequence of Lojasiewicz’s inequality (see [@Lo], §4.7). We prove below that $\psi \in L^p(\mu)$ for any $\psi \in \UU$ and $1 \leq p < + \infty$. Actually, we establish an estimate for $\int_{\NN_\psi[r]}  \abs{ \psi }^p$, useful to prove theorem B. We recall that $\mu \left( \NN_\psi[r]  \right) \leq c \, r^{\gamma}$ for some $c,{\gamma}> 0$ (see subsection \[consa\]).

\[cuun\] Let $\psi \in \UU$ and $1 \leq p < + \infty$. There exists $\kappa > 0$ such that: $$\label{dec}
\forall \, 0 < r < 1/2 \ \ 