• Bulletin of the Korean Mathematical Society
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• v.51 no.4
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• pp.965-978
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• 2014

In this paper we establish sharp $L^p$-regularity estimates for averaging operators with convolution kernel associated to hypersurfaces in $\mathbb{R}^d(d{\geq}2)$ of the form $y{\mapsto}(y,{\gamma}(y))$ where $y{\in}\mathbb{R}^{d-1}$ and ${\gamma}(y)={\sum}^{d-1}_{i=1}{\pm}{\mid}y_i{\mid}^{m_i}$ with $2{\leq}m_1{\leq}{\cdots}{\leq}m_{d-1}$.

• Journal of the Korean Mathematical Society
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• v.55 no.2
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• pp.373-390
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• 2018

In this paper we study the small-data scattering of the d dimensional fractional $Schr{\ddot{o}}dinger$ equations with d = 2, 3, $L{\acute{e}}vy$ index 1 < ${\alpha}$ < 2 and Hartree type nonlinearity $F(u)={\mu}({\mid}x{\mid}^{-{\gamma}}{\ast}{\mid}u{\mid}^2)u$ with max(${\alpha}$, ${\frac{2d}{2d-1}}$) < ${\gamma}{\leq}2$, ${\gamma}$ < d. This equation is scaling-critical in ${\dot{H}}^{s_c}$, $s_c={\frac{{\gamma}-{\alpha}}{2}}$. We show that the solution scatters in $H^{s,1}$ for any s > $s_c$, where $H^{s,1}$ is a space of Sobolev type taking in angular regularity with norm defined by ${\parallel}{\varphi}{\parallel}_{H^{s,1}}={\parallel}{\varphi}{\parallel}_{H^s}+{\parallel}{\nabla}_{{\mathbb{S}}{\varphi}}{\parallel}_{H^s}$. For this purpose we use the recently developed Strichartz estimate which is $L^2$-averaged on the unit sphere ${\mathbb{S}}^{d-1}$ and utilize $U^p-V^p$ space argument.