论文标题
$ L_P $ - $ L_Q $ $ fourier乘数在本地紧凑型量子组上
$L_p$-$L_q$ Fourier multipliers on locally compact quantum groups
论文作者
论文摘要
令$ \ mathbb {g} $为本地紧凑的量子组，带有dual $ \ wideHat {\ mathbb {g}} $。假设左HAAR重量$φ$和双重左HAAR重量$ \widehatφ$是奇特的，例如$ \ mathbb {g} $是一个单模型的kac代数。我们证明，对于$ 1 <p \ le 2 \ le q <\ infty $，傅立叶乘数$ m_ {x} $从$ l_p（\ widehat {\ widehat {\ sathbb {g}}，\wideHatφ）$ to $ l_q（\ widehat）to $ l_q（\ wide）在$ l_ {r，\ infty}（\ mathbb {g}，φ）$中，其中$ 1/r = 1/p-1/q $。而且，我们有 \ begin {equation*} \ | m_ {x}：l_p（\ wideHat {\ mathbb {g}}，\wideHatφ）\ to l_q（\ wideHat {\ sathbb {g}}，\wideHatφ） \ | x \ | _ {l_ {r，\ infty}（\ mathbb {g}，φ）}， \ end {equation*} 其中$ c_ {p，q} $仅取决于$ p $和$ q $。 Hörmander\ cite {Hormander1960}首先证明了这一点，以$ \ MATHBB {r}^n $证明，最近扩展到了更多的一般组和量子组。我们的工作涵盖了所有这些结果，证明更简单。特别是，这也产生了一个$ L_P $ - 倍数的乘数，而不是离散组von Neumann代数。还证明了$ \ Mathcal {s} _p $ - $ \ Mathcal {s} _Q $ Schur乘数的类似结果。
Let $\mathbb{G}$ be a locally compact quantum group with dual $\widehat{\mathbb{G}}$. Suppose that the left Haar weight $φ$ and the dual left Haar weight $\widehatφ$ are tracial, e.g. $\mathbb{G}$ is a unimodular Kac algebra. We prove that for $1<p\le 2 \le q<\infty$, the Fourier multiplier $m_{x}$ is bounded from $L_p(\widehat{\mathbb{G}},\widehatφ)$ to $L_q(\widehat{\mathbb{G}},\widehatφ)$ whenever the symbol $x$ lies in $L_{r,\infty}(\mathbb{G},φ)$, where $1/r=1/p-1/q$. Moreover, we have \begin{equation*} \|m_{x}:L_p(\widehat{\mathbb{G}},\widehatφ)\to L_q(\widehat{\mathbb{G}},\widehatφ)\|\le c_{p,q} \|x\|_{L_{r,\infty}(\mathbb{G},φ)}, \end{equation*} where $c_{p,q}$ is a constant depending only on $p$ and $q$. This was first proved by Hörmander \cite{Hormander1960} for $\mathbb{R}^n$, and was recently extended to more general groups and quantum groups. Our work covers all these results and the proof is simpler. In particular, this also yields a family of $L_p$-Fourier multipliers over discrete group von Neumann algebras. A similar result for $\mathcal{S}_p$-$\mathcal{S}_q$ Schur multipliers is also proved.
