  On $\mathbb{R}^n,$ a classical result due to Bourgain establishes the
restricted weak $(\frac{n}{n-1},1)$ inequality for the full maximal function
$M_F^{d\sigma}$ associated to the spherical averages. In this work we present
an extension to Bourgain's result on graded Lie groups for a family of full
maximal operators. We formulate this extension using the group Fourier
transform of the measures under consideration and the symbols of (positive
Rockland operators which are) positive left-invariant hypoelliptic partial
differential operators on graded Lie groups.
