偏置超立方体上基于非迹婴儿Fock模型的维数无关Riesz变换估计
Dimension-free Riesz transform estimates on biased hypercubes via non-tracial baby Fock models
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中文总结 AI 辅助
本文通过非迹婴儿Fock模型转移，证明偏置超立方体上Riesz变换的维数无关Meyer不等式，并推广至q-Araki--Woods代数。
中文摘要 AI 辅助
我们研究配备非均匀乘积测度的超立方体上的维数无关Riesz变换估计。对于$2\leq p<\infty$，我们证明了相关的carré du champ的维数无关Meyer不等式，其常数与维数和乘积测度均无关。与均匀情形相比，carré-du-champ平方函数通常不与无权重Riesz平方函数一致。在权重参数的额外有界性假设下，我们进一步获得了后者的维数无关估计。我们的证明基于从偏置超立方体到非迹婴儿Fock模型的转移方法。我们在偏置超立方体上引入了新的湮灭算子和相关的Riesz变换，并推导出将它们与非迹婴儿Fock模型上的Riesz变换联系起来的显式公式。主要的分析成分是对所得非交换Riesz变换的维数无关$L_p$估计，该估计通过Pisier和Lust-Piquard风格的旋转论证证明。相同的分析机制也在$q$-Araki--Woods代数上产生了维数无关的Riesz变换估计。
英文摘要
We study dimension-free Riesz transform estimates on hypercubes equipped with non-uniform product measures. For $2\leq p<\infty$, we prove a dimension-free Meyer inequality for the associated carré du champ, with constants independent of both the dimension and the product measure. In contrast to the uniform case, the carré-du-champ square function does not in general coincide with the unweighted Riesz square function. Under an additional boundedness assumption on the weight parameters, we further obtain dimension-free estimates for the latter. Our proof is based on a transference from the biased hypercube to a non-tracial baby Fock model. We introduce new annihilation operators and the associated Riesz transforms on the biased hypercube, and derive explicit formulas relating them to the Riesz transforms on the non-tracial baby Fock model. The main analytic ingredient is a dimension-free $L_p$-estimate for the resulting noncommutative Riesz transform, proved by a rotation argument in the spirit of Pisier and Lust-Piquard. The same analytic mechanism also yields dimension-free Riesz transform estimates on $q$-Araki--Woods algebras.
发表机构
- Seoul National University（首尔大学）
- Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas（西班牙高等科学研究理事会数学科学研究所）
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