Quasiradial Fourier multipliers and their maximal operators (Q2731620)

Let \(\rho\in C^{[\frac n2+1]}({\mathbb{R}}^n\setminus\{0\})\) be a distance function which is homogeneous with respect to a dilation group \(\{\exp(P\log t)\}_{t>0}\), where \(P\) is a real \(n\times n\) matrix whose eigenvalues \(\alpha_i\) satisfy \(\text{ Re}(\alpha_i)>0\). For \(f\in{\mathcal S}({\mathbb{R}}^n)\), the author studies the maximal operators NEWLINE\[NEWLINE{\mathcal M}_{m\circ\rho}f(x)=\sup_{t>0} |[m\circ(\rho/t)\widehat f]^\vee(x)|,NEWLINE\]NEWLINE where \(m(s)\) is a function on \((0,\infty)\). Suppose that \(m\) vanishes at infinity and satisfies NEWLINE\[NEWLINE\int^\infty_0s^\delta|m^{(\delta+1)}(s)|ds\leq CNEWLINE\]NEWLINE for \(\delta>n(1/p-1/2)\) and \(1\leq p\leq 2\), where \(m^{(\delta+1)}\) is the fractional derivative of order \(\delta+1\) of \(m\). Then, \textit{H. Dappa} and \textit{W. Trebels} [Ark. Mat. 23, 241-259 (1985; Zbl 0584.42012)] proved that \({\mathcal M}_{m\circ\rho}\) is bounded on \(L^p({\mathbb{R}}^n)\) for \(1<p\leq\infty\) and is of weak type \((1,1)\). In this paper, the author gives a new and simple proof of this result based on the vector-valued Hörmander multiplier theorem. An application to the boundedness of some Littlewood-Paley \(g\)-function is also given.