Indefinite higher Riesz transforms

Toshiyuki Kobayashi Research Institute for Mathematical Sciences, Kyoto University Andreas Nilsson Saab AB, Saab Aerosystems, TDAA-AN

TBD mathscidoc:1701.333145

Arkiv for Matematik, 47, (2), 331-344, 2007.4
Stein’s higher Riesz transforms are translation invariant operators on$L$^{2}($R$^{$n$}) built from multipliers whose restrictions to the unit sphere are eigenfunctions of the Laplace–Beltrami operators. In this article, generalizing Stein’s higher Riesz transforms, we construct a family of translation invariant operators by using discrete series representations for hyperboloids associated to the indefinite quadratic form of signature ($p$,$q$). We prove that these operators extend to$L$^{$r$}-bounded operators for 1<$r$<∞ if the parameter of the discrete series representations is generic.
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@inproceedings{toshiyuki2007indefinite,
  title={Indefinite higher Riesz transforms},
  author={Toshiyuki Kobayashi, and Andreas Nilsson},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203624633494954},
  booktitle={Arkiv for Matematik},
  volume={47},
  number={2},
  pages={331-344},
  year={2007},
}
Toshiyuki Kobayashi, and Andreas Nilsson. Indefinite higher Riesz transforms. 2007. Vol. 47. In Arkiv for Matematik. pp.331-344. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203624633494954.
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