Let ${A_t)}_{t>0}$ be a dilation group given by $A_t = exp(-P log t)$, where P is a real $n imes n$ matrix whose eigenvalues has strictly positive real part. Let $ u$ be the trace of P and $P^*$ denote the adjoint of pp. Suppose that $K$ is a function defined on $R^n$ such that $$mid$K(x)$mid$ leq k($mid$x$mid$_Q)$</TEX> for a bounded and decreasing function $k(t) on R_+$ satisfying $k diamond $mid$cdot$mid$_Q in cup_{varepsilon >0}L^1((1 + $mid$x$mid$)^varepsilon dx)$</TEX> where $Q = int_{0}^{infty} exp(-tP^*) exp(-tP)$ dt and the norm $$mid$cdot$mid$_Q$</TEX> stands for $$mid$x$mid$_Q = sqrt{<Qx, x>}, x in R^n$</TEX>. For $f in L^1(R^n)$, define $mf(x) = sup_{t>0}$mid$K_t * f(x)$mid$$</TEX> where $K_t(X) = t^{- u}K(A_{1/t}^* x)$. Then we show that $m$ is a bounded operator of $L^1(R^n) into L^{1, infty}(R^n)$.