0
$\begingroup$

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.

$\endgroup$
3
  • 1
    $\begingroup$ 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 $m\ne n$).. $\endgroup$
    – fedja
    Sep 18, 2016 at 21:01
  • $\begingroup$ Yes, in that case I meant $|m|\neq |n|$. Thanks for the remark. $\endgroup$
    – scouser
    Sep 19, 2016 at 10:25
  • $\begingroup$ @fedja: Apparently someone did bother: see the list in my answer. :-) $\endgroup$ May 16, 2020 at 21:14

1 Answer 1

1
$\begingroup$

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

$\endgroup$
1
  • $\begingroup$ Wow, terrific bibliography! Thank you! $\endgroup$ May 16, 2020 at 21:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.