247A, Notes 3: Singular integral operators

— 2. Calderón–Zygmund theory —

In the Hilbert transform, we have an operator which is bounded on , and we wish to extend this boundedness to other spaces as well. This is of course not automatic – the “enemy” is that the operator might map a broad shallow function into a tall narrow function of comparable norm (which will increase the norms markedly for all ), or conversely map a tall narrow function down to a broad shallow function of comparable norm (which will increase the norms markedly for ). It turns out that these two phenomena are more or less dual to each other, and so when working with an operator which is “comparable” to its adjoint (which is for instance the case with the skew-adjoint Hilbert transform, ) eliminating one of these enemies will automatically remove the other.

For operators such as the Hilbert transform, there are at least two approaches to exclude these scenarios. One is to exploit the phenomenon that if is not too large locally, then also doesn’t fluctuate too much locally; this prevents the “broad shallow to tall narrow” enemy which blocks boundedness for large . The dual approach is to show that if is localized and fluctuating, then is also localized; this prevents the “tall narrow to broad shallow” enemy which blocks boundedness for small . (There are also more algebraic methods, relying on explicit formulae for things like the norm of , or of the size of level sets of for various sets , but we will not discuss those here as they do not extend well to the more general class of singular kernels considered here.)

We shall first discuss the latter approach, which is the traditional approach to the subject, and return to the former approach later. We shall generalize from the Hilbert transform to a useful wider class of operators, which are integral operators “away from the diagonal”. To motivate this concept, observe that if is compactly supported and lies outside of the support of , then