Journal of Functional Analysis ( IF 1.7 ) Pub Date : 2020-05-23 , DOI: 10.1016/j.jfa.2020.108652 Yaryong Heo , Sunggeum Hong , Chan Woo Yang
In this paper we study natural generalizations of the first order Calderón commutator in higher dimensions . We study the bilinear operator which is given by Our results are obtained under two different conditions of the multiplier m. The first result is that when is a regular Calderón-Zygmund convolution kernel of regularity , maps into for all , as long as . The second result is that when the multiplier satisfies the Hörmander derivative conditions for all , and for all multi-indices α with , maps into for all , as long as . These two results are sharp except for the endpoint case . In case and , it is well-known that maps into for , as long as . In higher dimensional case , in 2016, when is the Riesz multiplier on , P. W. Fong, in his Ph.D. Thesis [9], obtained for as long as . As far as we know, except for this special case, there has been no general results for the off-diagonal case in higher dimensions . To establish our results we develop ideas of C. Muscalu and W. Schlag [18], [19] with new methods.
中文翻译:
高维一阶换向器的非对角线估计
在本文中,我们研究高阶一阶Calderón换向器的自然概括 。我们研究双线性算子 由我们的结果是在乘数m的两个不同条件下获得的。第一个结果是,当 是正则的常规Calderón-Zygmund卷积核 , 地图 进入 对所有人 , 只要 。第二个结果是当乘数 满足Hörmander导数条件 对所有人 ,并为所有多指数α与, 地图 进入 对所有人 , 只要 。除端点情况外,这两个结果都很清晰。以防万一 和 众所周知 地图 进入 对于 , 只要 。在高维情况下,在2016年, 是Riesz乘数 ,PW Fong,拥有博士学位 论文[9],获得 对于 只要 。据我们所知,除了这种特殊情况外,对角线情况没有任何一般结果 在更高的尺寸 。为了建立我们的结果,我们用新方法开发了C. Muscalu和W. Schlag [18],[19]的想法。