# Discrete Calderon-Zygmund operators

I would like to know whether there exists a Calderon-Zygmund theory discrete singular kernels. In particular I am interested when the discrete operator $T$ with kernel $K(n,m)$ given by $$(Tf)(n)=\sum_{m\neq n}K(n,m)f(m)=\sum_{m\neq n} \frac{m \,\chi(|m|\leq M)}{n^2-m^2}f(m)$$ with $m,n\in \mathbb{Z}$ is bounded as an operator from $\ell^2 \to \ell^2$.

Even more generally, maybe there exists a theorem for even more general discrete kernels, say $$K(n,m)=\frac{\lambda'(m)}{\lambda(n)-\lambda(m)}$$ for some function $\lambda: \mathbb{Z}\to \mathbb{R}$.

References are welcome.

• Same story, same theorems, same proofs (though I have no idea if anyone ever bothered to write them down in a textbook). In your particular case, just combine $\chi$ with $f$ and notice that the kernel is the difference $\frac1{n-m}-\frac{1}{n+m}$, so it is just the classical Hilbert transform in disguise (I hope, of course, that you meant $|m|\ne|n|$, not just $m\ne n$).. Sep 18, 2016 at 21:01
• Yes, in that case I meant $|m|\neq |n|$. Thanks for the remark. Sep 19, 2016 at 10:25
• @fedja: Apparently someone did bother: see the list in my answer. :-) May 16, 2020 at 21:14

Fedja's comment applies perfectly to your situation, but if you are interested in the general picture, here are some references. I am no expert, but I happen to have a paper with Rodrigo Bañuelos, where he wrote in the introduction:

The literature on harmonic analysis in the discrete setting, and in particular on singular integrals and other classical operators, has a long history. In addition to the papers listed above, a good sample of this literature can be found in works by Calderón and Zygmund [10]; Pierce [33]; Lust-Piquard [28]; Stein and Wainger [36], [37]; and Magyar, Stein, and Wainger [29].

Here are the references:

[10] A. P. CALDERÓN and A. ZYGMUND, On the existence of certain singular integrals, Acta Math. 88 (1952), 85–139. MR 0052553. DOI 10.1007/BF02392130

[33] L. B. PIERCE, Discrete analogues in harmonic analysis, Ph.D. dissertation, Princeton University, Princeton, 2009. MR 2713096

[28] F. LUST-PIQUARD, Dimension free estimates for discrete Riesz transforms on products of abelian groups, Adv. Math. 185 (2004), no. 2, 289–327. MR 2060471. DOI 10.1016/j.aim.2003.07.002

[36] E. M. STEIN and S. WAINGER, Discrete analogues in harmonic analysis, I: $$\ell^2$$ estimates for singular Radon transforms, Amer. J. Math 121 (1999), no. 6, 1291–1336. MR 1719802

[37] E. M. STEIN and S. WAINGER, Discrete analogues in harmonic analysis, II: Fractional integration, J. Anal. Math 80 (2000), 335–355. MR 1771530. DOI 10.1007/BF02791541

[29] A. MAGYAR, E. M. STEIN, and S. WAINGER, Discrete analogues in harmonic analysis: Spherical averages, Ann. of Math. (2) 155 (2002), no. 1, 189–208. MR 1888798. DOI 10.2307/3062154

• Wow, terrific bibliography! Thank you! May 16, 2020 at 21:51