Maximal semigroup symmetry and discrete Riesz transforms (Q2800022)

It is well known that the \textit{Riesz transform} \(R=(R_1,\dots,R_n)\) on \(\mathbb{R}^n\) is, up to a multiplicative constant, the uniquely determined \(n\)-tuple of \(L^2(\mathbb{R}^n)\)-bounded, translation and dilation invariant operators satisfying NEWLINE\[NEWLINE l_{g^{-1}}\circ R_j\circ l_g=\sum_{k=1}^n g_{jk}R_k, \qquad j=1,\dots,n, NEWLINE\]NEWLINE for every \(g=[g_{jk}]\in \mathrm{SO}(n,\mathbb{R})\). Here \(l_g\) denotes the action of \(g\) on \(L^2(\mathbb{R}^n)\), \(l_gf=f\circ g^{-1}\). In dimension one \textit{R. E. Edwards} and \textit{G. I. Gaudry} [Littlewood-Paley and multiplier theory. York: Springer-Verlag (1977; Zbl 0467.42001)] proved a similar characterization for the \textit{Hilbert transform} on \(\mathbb{T}\) and \(\mathbb{Z}\). The main result of the reviewed paper extends this characterization to the multidimensional setting. It is proved (Theorem A) that for the Riesz transforms naturally defined on \(\mathbb{T}^n\) and \(\mathbb{Z}^n\), respectively, a characterization analogous to that on \(\mathbb{R}^n\) holds. The result is formulated in a slightly abstract but adequate form using the notion of the so called \textit{maximal semigroup property} that an \(n\)-tuple of operators is supposed to satisfy. Moreover, it is also shown (Theorem B) that unlike the \(\mathbb{R}^n\) case, there exist infinitely many linearly independent multipliers on \(\mathbb{T}^n\) and \(\mathbb{Z}^n\) respectively, that enjoy the same maximal semigroup property as the Riesz transforms on \(\mathbb{T}^n\) and \(\mathbb{Z}^n\), provided that \(n\geq3\) and is not a multiple of four.