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We consider the problem of estimating the integral of the square of a densityf from the observation of a n sample. Our method to estimate $\int_{\mathbb{R}} f^2(x){\rm d}x$ 
isbased on model selection via some penalized criterion. We prove that our estimator achieves the adaptive rates established by Efroimovich and Low on classes of smooth functions. A key point of the proof is an exponentialinequality for U-statistics of order 2 due to Houdré and Reynaud.
