On the sign of the Green’s function associated to Hill’s equation with an indefinite potential

In this note we give a LpLp-criterium for the positiveness of the Green’s function of the periodic boundary value problem:x″+a(t)x=0,x(0)=x(T),x′(0)=x′(T)with and indefinite potential a(t)a(t). Moreover, we prove that such Green’s function is negative provided a(t)a(t) belongs to the image of a suitable periodic Ricatti type operator.