On the harmonic and geometric maximal operators

We examine the harmonic and geometric maximal operators defined for a general basis of open sets in ${\mathbb{R}}^n$. We prove two weight norm inequalities for the harmonic maximal operator assuming testing conditions over characteristic functions of unions of sets from the basis. We also prove a that a bumped two weight $A_p$-like condition is sufficient for the two weight boundedness of the harmonic maximal operator.