Maximal functions related to subelliptic operators invariant under an action of a nilpotent Lie group

On the domain ${\displaystyle {\Omega }_a=\{(x,b):x\in N,b\in {ℝ}^{+},b>a\}}$, where N is a simply connected nilpotent Lie group and a ≥ 0, certain N-invariant second order subelliptic operators L are considered. Every bounded L-harmonic function F is the Poisson integral ${\displaystyle F(x,b)=f\ast \mu {̌}_a^b\left(x\right)}$ for an ${\displaystyle f\in L^{\infty }\left(N\right)}$. The main theorem of the paper asserts that under some assumptions the maximal functions ${\displaystyle M_{1}f\left(x\right)={\supset }_{b\ge a+1}|f\ast \mu {̌}_a^b\left(x\right)|}$, !$!M_2f(x) = sup_{a