The Coltar-Stein lemma is a powerful tool to deal with boundedness of some translation-invariant operators such as convolution operators, which can be expressed as

I recently began to understand how powerful it might be via the method. I would like to reproduce its proof following Fefferman’s presentation.

Suppose is a sum of operators on Hilbert spaces (say the usual ). Assume that

and . Then .

Here is a non-negative even function.

The argument follows an idea of iteration.

Step 1. . We estimate into two ways. Firstly it is easy to see that

; secondly it is also trival that

. Hence by taking the geometric means of these two bounds, we have

.

Hence

.

This gives that

Step 2. We investigate one more iteration of . We write

Also we estimate by organizing the operators in two ways.

Firstly since , we see that

secondly since ,

By taking geometric means, we see that

Hence

.

Step 3. by induction, we see that for any , we have

.

Let , we see that

. The proof of this lemma is complete.

We remark that in the argument both bounds and are used.

Let where be the Littlerwood-Paley projection operator. It is easy to see that is a self-adjoint operator. Also and for .

Then Stein-Cotlar gives,

,

which matches that .

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