Let $\varphi$ be a plurisubharmonic function in the unit ball $B\_1\subset \mathbb{C}^n$ with $\varphi\le 0$. Suppose that the Lelong number $\nu(\varphi,0)<k$ for some $k>0$. Does it follow that there exists $\alpha>0$, possibly depending on $k$, such that $\int\_{B\_{\frac{1}{2}}}e^{-\alpha\varphi}dvol(z)\le C$?


Note that this does not immediately follow from the definition of Lelong number. Since $\nu(\varphi,0)=\lim\inf\_{z\rightarrow0}\frac{\varphi(z)}{\log|z|}$, so $\nu(\varphi,0)<k$ apriori only implies $\varphi(z)-k\log|z|$ bounded from below along some sequence $z\_i\rightarrow0$.


On the other hand, if we know $\nu(\varphi,0)=0$, then above is indeed true, since under this assumption, we would have $\varphi(0)$ bounded from below, and the result above follows from a Lemma of Hormander(Lemma 4.4 in his book on several complex variables).



Lelong numbers and integrability of psh functions
**The answer is yes** :


If the Lelong number of a plurisubharmonic function
$\varphi$ at a point $a$ satisfies the condition $\nu(\varphi,a) < 2$, then the function $e^{-\varphi}$ is locally
integrable with respect to the Lebesgue measure in a neighbourhood of $a$.


It follows from a result of Skoda, Proposition 7.1 in



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> H. Skoda, [Sous-ensembles analytiques d’ordre fini ou infini dans $\mathbb{C}^n$](https://eudml.org/doc/87191), Bull. Soc. Math. de France 100 (1972), 353-408.
> 
> 
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