Classical Gårding inequalities such as those of Hörmander, Hörmander-Melin or Fefferman-Phong are proved by pseudo-differential methods which do not allow to keep a good control on the supports of the functions under study nor on the smoothness of the coefficients of the operator. In this paper, we show by very simple calculations that in certain special situations, the results that can be obtained directly are much better than those expected thanks to the general theory.