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.
© Toshiyuki Kobayashi