Lecture 3: Convolution-type Singular Integral Operators

We next turn to the study of singular integral operators. The canonical example to keep in mind is the Hilbert transform, which is given by
Using the Fourier Transform it is easy to see that the Hilbert transform is bounded on $L^2(R)$. Then using connections to complex analysis it follows relatively easily that it is in fact bounded on all $L^p(R)$.

In this lecture we will study certain generalizations of the Hilbert transform. The kernel associated to the Hilbert transform, $\frac{1}{x}$, is just barely non square integrable on $R$, but instead possesses certain size and smoothness properties that are suitable replacements. We will consider relatively general convolution-type singular integral operators and show that the general behavior as indicated by the Hilbert transform in fact persists. This lecture will introduce fundamental tools in harmonic analysis: interpolation theorems and Calderon-Zygmund Decompositions.

Sorry, this was a little

Sorry, this was a little confusing.

If we assume that the kernel K satisfies the conditions in Theorem 1.1, namely size, smoothness and cancellation, then we can show that the truncations of the kernel have uniformly bounded Fourier transform, independent of the truncation $\epsilon$. The idea is carried out now in Lecture 4 where we study a simple case of multi-parameter singular integral operators.

Next, if we assume that the kernel $K$ satisfies the conditions of Theorem 1.1, then it is possible to show that $K_\epsilon$ also satisfies the same estimates as the original kernel $K$, except we may have a slightly worse constant depending on the dimension. To see this, one should first consider the kernel $K_1$ and show that one has the same estimates as the original kernel. Then one should recognize that the kernels in question have a certain dilation invariance that allows one to reduce the case of general $K_\epsilon$.

Here, unfortunately, I had a reference to the wrong theorem, and I think this is where the confusion may have arisen. I have corrected the lecture, and posted an update. Please let me know if this doesn't clarify things for you.


cut-off kernel

Hi Brett,

Can you please explain what one needs to prove about the cut-off kernel K_epsilon? So initially K satisfies the condition from Theorem 1.1, but I don't understand exactly what we have to show. And then you say that K_epsilon satisfies the kernel conditions from Theorem 1.2. For this one, you assume that the initial kernel K satisfies the conditions from Theorem 1.1 or 1.2?

Thank you,