Journal of the
Korean Mathematical Society JKMS

Let $\{A_t\}_{t>0}$ be a dilation group given by $A_t=\exp (-P$ $\log t)$, where $P$ is a real $n\times n$ matrix whose eigenvalues has strictly positive real part. Let $\nu$ be the trace of $P$ and $P^*$ denote the adjoint of $P$. Suppose that $\Cal K$ is a function defined on ${\Bbb R}^n$ such that $|\Cal K (x)|\le \frak K (|x|_Q)$ for a bounded and decreasing function $\frak K (t)$ on ${\Bbb R}_+$ satisfying $\frak K\circ |\cdot |_Q \in \cup_{\varepsilon>0} L^1 ((1+|x|)^{\varepsilon}dx)$ where $Q=\int_0^{\infty}\exp (-t P^*)\exp (-t P)dt$ and the norm $|\cdot |_Q$ stands for $|x|_Q=\sqrt {\langle Q x, x \rangle }, x\in {\Bbb R}^n$. For $f\in L^1({\Bbb R}^n)$, define $\frak M f(x)=\sup_{t>0} |{\Cal K}_t *f(x)|$ where ${\Cal K}_t (x)=t^{-\nu}\Cal K (A^*_{1/t} x)$. Then we show that $\frak M$ is a bounded operator of $L^1 ({\Bbb R}^n)$ into $L^{1,\infty} ({\Bbb R}^n)$.

Keywords: maximal operator, weak type $L^1 ({\Bbb R}^n)$-estimate, quasiradial majorant, maximal weighted $L^1$-space, maximal Bochner-Riesz operator

MSC numbers: 42B25, 42B15