## T. Kobayashi, A. Nilsson, and F. Sato.
Maximal semigroup symmetry and discrete Riesz transforms.
Journal of the Australian Mathematical Society **100** (2016), issue 2, 216-240, Published Online, 14 December 2015. DOI: 10.1017/S144678871500049X..

We raise a question
if the Riesz transform on **T**^n
or **Z**^n is characterized by the
''maximal semigroup symmetry that they satisfy?
We prove that this is the case if and only if the dimension
*n=1,2* or a multiple of four.
This generalizes a theorem of Edwards and Gaudry for the Hilbert
transform (*i.e.* the *n=1* case) on **T** and **Z**,
and extends a theorem of Stein for the Riesz transform on
**R**^n.
Unlike the **R**^n case,
we show that there exist infinitely many, linearly independent
multiplier operators that enjoy the same maximal semigroup symmetry
as the Riesz transforms on **T**^n and
**Z**^n if *n>=3* and is not a multiple of four.

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© Toshiyuki Kobayashi