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?
Best Answer
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).