论文标题
三角混乱和$ \ mathrm {x} _p $不平等i-离散组平衡的傅立叶截断
Trigonometric chaos and $\mathrm{X}_p$ inequalities I -- Balanced Fourier truncations over discrete groups
论文作者
论文摘要
我们研究了对它们的差异操作员，研究组代数的傅立叶截断平均截断平均值的$ l_p $估计值。我们的结果将NAOR的基本不平等扩展到了离散组的HyperCube（在度量几何形状中产生的深远影响）。根据方向衍生物的方向衍生物建立了不同的不平等，这些衍生物是通过傅立叶截断确定的仿射表示构建的。我们的证明依赖于非交易性$ l_p $ - 空格的Banach $ \ Mathrm {x} _p $的性质，而无需riesz变换的无维估计。在自由组的特殊情况下，我们使用基于自由希尔伯特变换的替代方法。
We investigate $L_p$-estimates for balanced averages of Fourier truncations in group algebras, in terms of differential operators acting on them. Our results extend a fundamental inequality of Naor for the hypercube (with profound consequences in metric geometry) to discrete groups. Different inequalities are established in terms of directional derivatives which are constructed via affine representations determined by the Fourier truncations. Our proofs rely on the Banach $\mathrm{X}_p$ nature of noncommutative $L_p$-spaces and dimension-free estimates for noncommutative Riesz transforms. In the particular case of free groups we use an alternative approach based on free Hilbert transforms.
