The write-up, as always, is here, while the motivation and added exposition is below.
There is the following result from the first-generation Calderón-Zygmund theory:
Theorem [1, Proposition 5.4.4, p. 379]: Suppose \(W \in \mathcal{S}'(\mathbb{R}^d)\) is given by a function \(K : \mathbb{R}^d \setminus \{0\} \to \mathbb{C}\), \(K \in L^1_{\text{loc}}(\mathbb{R}^d \setminus \{0\})\), when restricted to the open set \(\mathbb{R}^d \setminus \{0\}\). Suppose further that \(K\) obeys a bound $$\int_{r \leq |x| \leq 2 r}|K(x)| \, dx \leq A$$ uniformly for all choices of \(r > 0\). Suppose the distribution-valued operator $$T : \mathcal{S}(\mathbb{R}^d) \to \mathcal{S}'(\mathbb{R}^d)$$ given by convolution against \(W\) (\(T(f) = W * f\)) actually maps into \(L^2(\mathbb{R}^d)\), and is is \(L^2\)-bounded.
Then $$\sup_{0 < r < R < \infty} \left| \int_{r \leq |x| \leq R} K(x) \, dx \right| < \infty.$$
The proof goes something like this: by a result of Stein on classification of translation-invariant operators on \(L^2\), this is given by convolution with a tempered distribution whose Fourier transform is a bounded (or, rather, \(L^{\infty}\)) function. One then writes integrals like $$\int_{\mathbb{R}^d} K(x) (\phi(x / R) \ – \, \phi(x / r)) \, dx$$ for bump functions \(\phi\), adapted the unit ball, and then one is supposed to see this equals $$(W * \delta_R \phi)(0) \ – \, (W * \delta_r \phi)(0) = \langle \widehat{W}, \widehat{\delta_R \phi} \rangle \ – \, \langle \widehat{W}, \widehat{\delta_r \phi} \rangle,$$ using the tempered distribution representation. One then uses that the Fourier transform of the dilated-bump is $$R^d \widehat{\phi}(R \xi).$$ After using the annulus-type condition on \(K\) to treat some of the physical-space spillover regions, and playing with some of these types of integrals, one gets integrals like $$\int_{\mathbb{R}^d} \widehat{W}(\xi) R^d \widehat{\phi}(R \xi) \, d\xi$$ for \(r\), \(R\), which are bounded in magnitude by \(\lesssim_d A + \|\widehat{W}\|_{L^{\infty}(\mathbb{R}^d)} \|\widehat{\phi}\|_{L^1(\mathbb{R}^d)}\).
I do not like this theorem, even though it does show that the cancellation is close to optimal, in a sense.1 First, the reliance on the auxiliary result of classification of the translation-invariant bounded linear operators on \(L^2\) (that they are multipliers) is somewhat messy. This is not an especially-interesting question to me; it is arguably more algebraic than analytic. In general, I like results that introduce tools which can be used in other contexts; the proof of this result feels somewhat full-of-surprises: there are plenty of moves that show up out of nowhere, and feel bafflingly-difficult to stumble upon naturally (honestly, I’ve never quite understood the argument on an intuitive level).
Furthermore, the proof of this result itself is somewhat irritating, because when we take the representation of the integral as \((W * \phi)(0) = (T (\phi))(0)\), we are doing the one thing that should almost never be done with an \(L^p\) function: we are evaluating it pointwise. Every analyst, when they’re first introduced to \(L^p\) spaces, is repeatedly reminded how pointwise evaluation is violently-discontinuous. It is drilled into our minds that pointwise representations are generally unsafe, and we should only be evaluating functions in a suitable-protected way, through a well-defined duality pairing with some test function. We go through our schooling learning about using normed and seminormed quantities, various tricks for averaging and integrating, ways to extract information from behavior in distribution, or other weak forms of probing, all in an attempt to avoid ever handling a function’s pointwise value. Seeing an argument that relies so heavily on this forbidden operation is rather unsettling, to say the least. I have no idea who came up with this argument, or if this is the original method they used, but it would be nice to see a proof of this that did not rely on the device of evaluating an \(L^p\) function at a distinguished point.
Continuing in this vein, there is a result of Stein speaking directly to this, which I find much more pleasing, though it uses a bit of (what some might consider) high-powered machinery:
Theorem [2, Chapter VIII, Theorem 4, p. 306]. Let \(K : (\mathbb{R}^d \times \mathbb{R}^d)^* \to \mathbb{C}\) be a standard kernel. Then there exists a Calderón-Zygmund operator \(T\) with associated kernel \(K\) iff, for the functions defined by $$I_{\epsilon, N}(x) = \int_{\epsilon < |x – y| < N} K(x, y) \, dy, \qquad I_{\epsilon, N}^*(y) = \int_{\epsilon < |y – x| < N} K(x, y) \, dx$$ for \(0 < \epsilon < N < \infty\), we have the properties of boundedness in mean $$\sup_{x_0 \in \mathbb{R}^d} \int_{B(x_0, N)} |I_{\epsilon, N}(x)| \, dx \leq C \cdot N^d, \qquad \sup_{y_0 \in \mathbb{R}^d} \int_{B(y_0, N)} |I_{\epsilon, N}^*(y)| \, dy \leq C \cdot N^d$$ for some uniform constant \(C \geq 0\), over all pairs of \((\epsilon, N)\).
(Stein gives this as an \(L^2\) average, as his formulation of the \(T(1)\) theorem is stated for normalized bumps which are mapped to elements of \(L^2\) with particular bounds. However, using (a corollary of) Tao’s \(T(1)\) theorem, discussed previously on this blog, we see that it is enough to use the weakest average: the \(L^1\) average, and so I have taken the liberty to recast the statement with this weaker hypothesis.)
Because the ranges of integration are each over bounded, well-controlled sets, we get that this is something like \(T_{\epsilon, N}(\chi_{B(x_0, 2 N)})\) inside each integral, and so this is really an integral that can be handled by the \(L^2\) boundedness of the \(T_{\epsilon, N}\). That boundedness, of course, comes from the Cotlar’s inequality for maximal truncations of third-generation Calderón-Zygmund operators. Consequently, we get that this condition is actually also necessary if \(K\) is to be associated with any genuine CZO.
Conversely, if this condition is met, then we can use a few quick inequalities to show that the conditions for (the corollary to) Tao’s \(T(1)\) theorem are satisfied, and hence that each \(T_{\epsilon, N}\) has \(\lesssim_{d, K, I} 1\) operator norm on \(L^2(\mathbb{R}^d)\), independent of the truncation parameters \(\epsilon\) and \(N\). Consequently, taking a weak limit, we get the existence of a Calderón-Zygmund operator, whose singular kernel is precisely \(K\).
I find this approach (one direction with the Cotlar’s inequality, which is the natural tool for understanding truncations of a singular integral, and the other with the \(T(1)\) theorem, which is the natural tool whenever you have “diagonal-type” testing conditions of the form \(\int_{\delta B} |T_{\epsilon, N}(\chi_B)| \, d\lambda^d \lesssim \lambda^d(B)\), uniformly in balls \(B \subseteq \mathbb{R}^d\)), to be quite clear: an intuitive proof in both directions.
Even though this only works in the class of standard kernels, which means this is not a complete generalization of the above Theorem (it excludes the rough kernels of convolution type), I still found this to a gem of a theorem, and arguably the most natural optimal cancellation condition.2 Working out the details of this was a very fun exercise.
[1] Loukas Grafakos, Classical Fourier Analysis, 3d ed., Graduate Texts in Mathematics, vol. 249, Springer, New York, 2014. MR 3243734.
[2] Elias M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, N.J., 1993. MR 1232192.
Work begun: September 11, 2025. Completed: September 11, 2025.
- If we’re going to be precise about it, the if-and-only-if holds in the category of operators with kernels satisfying the size and Hörmander conditions. That is, for a kernel satisfying the size condition and Hörmander condition, the cancellation condition will be enough to obtain the existence of a Calderón-Zygmund operator whose associated kernel is \(K\); meanwhile, the result above shows that all Calderón-Zygmund operators whose associated kernels have those two (relatively unrestrictive) properties will also have \(K = K_T\) enjoying the cancellation condition. ↩︎
- Amusingly, this result also provides a means of verifying the \(L^2\) boundedness of the Hilbert transform (and, in higher dimensions, the Riesz transforms). The proof is, in fact, noncircular; providing one is willing to go through the trouble of learning about Carleson measures, \(\text{BMO}\), and many other things, Tao’s proof furnishes the requisite \(T(1)\) theorem independent of any theory of the Hilbert or Riesz transform(s), and from there, Stein’s cancellation theorem quickly follows. Of course, no one would seriously attempt to learn things in such an order; I have half a mind to add this proof to the famous “nuking mosquitoes” page on MathOverflow. ↩︎
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