Journal of the
Korean Mathematical Society JKMS

Let $\varrho\in C^{\infty}\big({\mathbb R}^n\setminus \{0\}\big)$ be a distance function which is homogeneous with respect to a dilation group $\{t^P\}_{t>0}$. For $f\in \frak S ({\mathbb R}^n)$, we consider the maximal operators generated by quasiradial Fourier multiplier $\frak m \circ\varrho$ which is defined by
$${\mathcal M}_{\frak m\circ\varrho} f(x) =\sup_{t>0}\left|{\mathcal F}^{-1}\big[\frak m \circ (\varrho/t)\widehat f\, \big](x)\right|$$ where $\frak m$ is a function given on ${\mathbb R}_+$. Suppose that the sphere $\Sigma_{\varrho}\fallingdotseq
\{\xi\in {\mathbb R}^n|\,\varrho (\xi)=1\}$ satisfies a certain finite type condition and that $\frak m$ vanishes at infinity and satisfies $\int_0^{\infty} s^{\delta} |{\frak m}^{(\delta+1)} (s)|\,ds\le C$ for $\delta>(n-1)\big|1/p-1/2 \big|$, $1\le p\le\infty$. Then we prove that ${\mathcal M}_{\frak m\circ\varrho}$ is bounded on $L^p({\mathbb R}^n)$ for $1

Keywords: quasiradial Fourier multipliers, weak type $(1, \, 1)$-estimate

MSC numbers: 42B15, 42B25