论文标题
作用于非交换$ L_P $ - 空格的傅立叶和Schur乘数的多线性转移
Multilinear transference of Fourier and Schur multipliers acting on non-commutative $L_p$-spaces
论文作者
论文摘要
令$ g $为本地紧凑的单型组，让$ ϕ $成为$ g $上$ n $变量的某些功能。对于这样的$ ϕ $，可以将多线性傅立叶乘数关联，该乘法器可用于von Neumann Algebra组的非交换性$ l_p $ - 空格的一些$ n $倍产品。一个人还可以定义关联的Schur乘数，该乘数在Schatten类$ s_p（l_2（g））$的$ n $ fold产品上。我们将众所周知的转移结果从线性情况下概括为多线性情况。特别是，我们表明，所谓的“多重界限$（p_1，\ ldots，p_n）$ - norm'的多线性Schur乘数在上面是由相应的傅立叶乘数的乘数乘数乘法性界限，只要组可进行时，均具有平等性。此外，我们证明了双线性希尔伯特变换不是作为向量映射$ l_ {p_1}（\ Mathbb {r}，s_ {p_1}）\ times l_ {p_2}（p_2}}（\ mathbb {r mathbb {r}，s_ {p_2}，s_ {p_2}）\ right arow l_ {r l_ s_ {1}）$，每当$ p_1 $和$ p_2 $都这样，$ \ frac {1} {p_1} + \ frac {1} {p_2} = 1 $。某些Calderón-Zygmund型操作员也有类似的结果。这与非载体有价值的欧几里得案相反。
Let $G$ be a locally compact unimodular group, and let $ϕ$ be some function of $n$ variables on $G$. To such a $ϕ$, one can associate a multilinear Fourier multiplier, which acts on some $n$-fold product of the non-commutative $L_p$-spaces of the group von Neumann algebra. One may also define an associated Schur multiplier, which acts on an $n$-fold product of Schatten classes $S_p(L_2(G))$. We generalize well-known transference results from the linear case to the multilinear case. In particular, we show that the so-called `multiplicatively bounded $(p_1,\ldots,p_n)$-norm' of a multilinear Schur multiplier is bounded above by the corresponding multiplicatively bounded norm of the Fourier multiplier, with equality whenever the group is amenable. Further, we prove that the bilinear Hilbert transform is not bounded as a vector valued map $L_{p_1}(\mathbb{R}, S_{p_1}) \times L_{p_2}(\mathbb{R}, S_{p_2}) \rightarrow L_{1}(\mathbb{R}, S_{1})$, whenever $p_1$ and $p_2$ are such that $\frac{1}{p_1} + \frac{1}{p_2} = 1$. A similar result holds for certain Calderón-Zygmund type operators. This is in contrast to the non-vector valued Euclidean case.
