Some Examples of the Calderón-Zygmund operators

In this post we study various examples of ``singular integral operator''. Those included here are

• The Riesz transforms, which recover the Hessian of a Schwartz function from its Laplacian;
• Regular parametrix of constant coefficient linear (hypo)elliptic operators, and their distributional derivatives, which are responsible for the regularity property of their solutions;
• Littlewood Paley projections, which is a convenient tool of multiresolution analysis.

In the above examples the kernel of the operator is ``singular'', in the sense that it's not always locally integrable, yet the operator has nice boundedness properties, established by Calderón and Zymund, and others. Roughly speaking, that they can have such properties is due to the following two properties:

• The kernels have certain cancellation condition, or when it is a convolutional operator, the closely related property that their Fourier transform (i.e. corresponding multiplier) is bounded. In general it is the $L^2$ boundedness of the operator. This in some sense makes it possible to bound those functions that have big height but low oscillation.
• The kernels, and their derivatives, have "integrable decay'', or the closely related property that their multipliers are $ {C^{\infty}}$ functions that satisfy some differential inequalities corresponding to that decay. This in some sense makes it possible to bound those functions that have low height but big oscillation.

There is a more precise procedure to implement above intuitive argument, that is the famous Calderón-Zygmund decomposition, together with real interpolation. However, it should be noted that the argument only leads to a weak type bound in the endpoint case $L^{1}$ and $L^{\infty}$, and in that case more delicate nature of the functions (cf. the $\bf{H^{1}}$ and $BMO$) have to be considered. These boundedness properties will not be studied here, instead I want to present how these examples arise naturally in various questions, then I will identify them as such operators, and explain why studying them is important.

Schauder estimate

From known results in potential theory for the Poisson equation

$ \displaystyle \Delta u=f, $

Schauder was able to develop a linear elliptic theory in the framework of classical solutions, which will be presented in this article. The fundamental observation is that locally, equations

$ \displaystyle Lu=f $

under appropriate assumptions can be regarded as perturbations of the Poisson equation. Before commencing the analysis we will first develop some intuition with Hölder spaces, as well as some general strategy in dealing with estimate with Hölder-type quantities. This will be done in Section 2. The attack on this perturbation analysis will thus need to face the following strategical questions:

• what are the most essential problems that one can reduce to? 
• what kind of estimates does one need to make?*
• what assumptions does one need, so that the results from the potential theory (or, the constant coefficient case)  become useful?

The somewhat vague question* shall be partially answered in Section 5, given the guidance from the Banach's two famous theorems in linear functional analysis. However, the technique will shine at its full extent only after one grasps the ideas in Section 3, where the first and the third questions are addressed. The answers to these questions, though not always explicit stated, should constitute the basic philosophy in attacking similar problems.

In the exposition we have largely followed the book by Gilbarg and Trudinger. Materials for Hölder spaces are mainly collected from notes of Tao.