  The study of dispersive properties of Schr\"odinger operators with point
interactions is a fundamental tool for understanding the behavior of many body
quantum systems interacting with very short range potential, whose dynamics can
be approximated by non linear Schr\"odinger equations with singular
interactions. In this work we proved that, in the case of one point interaction
in $\mathbb{R}^3$, the perturbed Laplacian satisfies the same $L^p-L^q$
estimates of the free Laplacian in the smaller regime $q\in[2,3)$. These
estimates are implied by a recent result concerning the $L^p$ boundedness of
the wave operators for the perturbed Laplacian. Our approach, however, is more
direct and relatively simple, and could potentially be useful to prove optimal
weighted estimates also in the regime $q\geq 3$.
