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# $L^p-L^q$ estimates for convolutions with distribution kernels having singularities on the light cone

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 Title: $L^p-L^q$ estimates for convolutions with distribution kernels having singularities on the light cone Authors: Cho, Yonggeun Browse this author Kim, Youngcheol Browse this author Lee, Sanghyuk Browse this author Shim, Yongsun Browse this author Keywords: cone convolution estimates Issue Date: 2005 Journal Title: Hokkaido University Preprint Series in Mathematics Volume: 684 Start Page: 1 End Page: 17 Abstract: We study the convolution operator $T^z$ with the distribution kernel given by analytic continuation from the function $$\widetilde{K}^z(y,s,t)= \left\{\begin{array}{ll} (t^2-s^2-|y|^2)_+^z/\Gamma(z+1)\quad &\mbox{if}\quad t>0\\ 0 \quad&\mbox{if} \quad t\le 0\end{array}\right\}, \quad Re(z)>-1$$ where $(y,s,t)\in \mathbb R^{n-1}\times\mathbb R\times \mathbb R$. We obtain some improvement upon the previous known estimates for $T^z$. Then we extend the result of the cone multiplier of negative order on $\mathbb{R}^3$ \cite{lee1} to the case of general $\mathbb{R}^{n+1},\, n \ge 2$. Type: bulletin (article) URI: http://hdl.handle.net/2115/69489 Appears in Collections: 理学院・理学研究院 (Graduate School of Science / Faculty of Science) > Hokkaido University Preprint Series in Mathematics