This is the second part of the series on basic Littlewood-Paley theory, which has been extracted from some lecture notes I wrote for a masterclass. In this part we will prove the Littlewood-Paley inequalities, namely that for any it holds that

This time there are also plenty more exercises, some of which I think are fairly interesting (one of them is a theorem of Rudin in disguise).

Part I: frequency projections.

** 4. Smooth square function **

In this subsection we will consider a variant of the square function appearing at the right-hand side of () where we replace the frequency projections by better behaved ones.

Let denote a smooth function with the properties that is compactly supported in the intervals and is identically equal to on the intervals . We define the *smooth frequency projections* by stipulating

notice that the function is supported in and identically in . The reason why such projections are better behaved resides in the fact that the functions are now smooth, unlike the characteristic functions . Indeed, they are actually Schwartz functions and you can see by Fourier inversion formula that ; the convolution kernel is uniformly in and therefore the operator is trivially bounded for any by Young’s inequality, without having to resort to the boundedness of the Hilbert transform.

We will show that the following smooth analogue of (one half of) () is true (you can study the other half in Exercise 6).

Proposition 3Let denote the square functionThen for any we have that the inequality

We will give two proofs of this fact, to illustrate different techniques. We remark that the boundedness will depend on the smoothness and the support properties of only, and as such extends to a larger class of square functions.

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