It seems to be a commonplace in harmonic analysis that if some operator (say, Fourier multiplier) is bounded on $L^p(\mathbb{R}^n)$ then by transference the similar operator is also bounded on $L^p(\mathbb{T}^n)$ (the most common example of application of this general principle is Hilbert transform for $n=1$). There exist certain general theorems which allow us to transfer the results in the setting of $\mathbb{R}^n$ to the setting of $\mathbb{T}^n$, and vice versa (for example, such theorems may be found in Grafakos, "Classical Fourier Analysis").


Now, my question is -- can we do the same for the cyclic group $\mathbb{Z}\_n$ (with the standard Haar measure on it)? Say, for arbitrary number $m<n$ we may define the operator $\widehat{Tf}=\chi\_{[0,m]}\hat{f}$ on $L^p(\mathbb{Z}\_n)$. It seems to be not difficult to see directly that such operator is bounded and its norm does not depend on $n$ and $m$ but can we somehow derive it from the fact that Hilbert transform is bounded on $L^p(\mathbb{R})$ and $L^p(\mathbb{T})$? The intuition here is that for large $n$ the group $\mathbb{Z}\_n$ should resemble $\mathbb{Z}$ (or $\mathbb{T}$) but I do not know how to do a rigorous proof of such "transference".


After that, one may go further and prove the "Littlewood--Paley theorem for $\mathbb{Z}\_n$" (uniformly in $n$) and maybe certain multiplier theorems. Did anybody address such questions (with or without transference)? Or maybe these questions are trivial for some reason?



Fourier multipliers and transference on cyclic groups
Ok, I found the answer myself, so I'll post it here. It turned out that the results for multipliers should be transferred not from $L^p(\mathbb{T})$ but from $\ell^p(\mathbb{Z})$. That is, the boundedness of operators that figure in the question follows from the boundedness of the following operator on $\ell^p(\mathbb{Z})$: we take a sequence in $\ell^p(\mathbb{Z})$, consider its Fourier transform (which is a function on $\mathbb{T}$), multiply it by characteristic function of an arc and take the Fourier coefficients of the resulting function. The boundedness of such operator on $\ell^p(\mathbb{Z})$ is of course known --- it can be found in the book by Edwards and Gaudry "Littlewood--Paley and multiplier theory" (1977).


As for this transference, it is known, too (and not very difficult): it is written in an article "Transference methods in analysis" by Coifman and Weiss (in a more general context; to be more specific, Theorem 3.15 and Corollary 3.16 can be applied here).


