Bounds for a cone-type multiplier operator of negative index in \(\mathbb{R}^{3}\) (Q2861584)

Let \(\gamma : [-1,1] \rightarrow \mathbb{R}\) be a smooth function and let NEWLINE\[NEWLINE\Sigma_{\gamma} = \{ (\xi, \tau) \in \mathbb{R}^{2} \times \mathbb{R} : (\xi, \tau) = \lambda(t, \gamma(t), 1), t \in [-1,1], \lambda > 0 \}NEWLINE\]NEWLINE be the conic surface which is generated by \(\mathcal{C} = \{ (t, \gamma(t)) \in \mathbb{R}^{2} : t \in [-1,1]\}\). The Fourier multiplier operator \(S^{\alpha}\) on \(\mathbb{R}^{3}\) of order \(\alpha\) is defined by NEWLINE\[NEWLINE\widehat{S^{\alpha}f}(\xi, \tau) = \frac{\phi(\tau)}{\Gamma(\alpha + 1)} \, \chi(\xi) \, (\xi_{2} - \tau \gamma(\frac{\xi_1} {\tau}))_{+}^{\alpha} \widehat{f}(\xi, \tau), \quad (\xi, \tau) = (\xi_1, \xi_2, \tau) \in \mathbb{R}^{2} \times \mathbb{R},NEWLINE\]NEWLINE where \(\phi \in C_{0}^{\infty}(1,2)\) and \(\chi\) is a smooth function compactly supported around \((0, \gamma(0))\). When \(\Sigma_{\gamma}\) is a subset of the light cone, \(S^{\alpha}\) is the regular cone multiplier operator. For this operator, when \(\alpha > 0\), the problem of \(L^p\) boundedness has been studied by several authors and the most recent result in this direction is due to \textit{G. Garrigós} and \textit{A. Seeger} [Proc. Edinb. Math. Soc., II. Ser. 52, No. 3, 631--651 (2009; Zbl 1196.42010)], \textit{Y. Heo} [Indiana Univ. Math. J. 58, No. 3, 1187--1202 (2009; Zbl 1172.42004)] and \textit{Y. Heo} et al. [Acta Math. 206, No. 1, 55--92 (2011; Zbl 1219.42006)]. When \(\alpha < 0\), \textit{S. Lee} [Bull. Lond. Math. Soc. 35, No. 3, 373--390 (2003; Zbl 1031.42008)] proved sharp \(L^p - L^q\) estimates on \(\mathbb{R}^{3}\).NEWLINENEWLINEThe author considers the operator \(S^{\alpha}\) of negative order \(\alpha\) and obtains sharp estimates for \(L^p\) to \(L^q\) boundedness, when the conic surface \(\Sigma_{\gamma}\) is generated by a curve \(\mathcal{C}\) whose curvature vanishes at a single point. Here the range of \(p\) depends on the type of the curve \(\mathcal{C}\).