A toolkit

In this chapter we present the main ingredients for our work, and the machinery we are going to apply. Some parts of it are very brief while some other are very detailed, especially those concerned with the Hilbert transform and Hardy spaces. The point is that these parts contain facts and results that have not appeared together in one source. Most of them are known but apparently do not play the same role they do in our study. Some others are not to be found in the literature at all. Such are, for instance, the example of an odd integrable function with nonintegrable Hilbert transform or the example of a function not belonging to the real Hardy space but with finite left-hand side in the Fourier–Hardy inequality (see Subsection 3.3.2 in Chapter 3). In any case, we hope that this collection will be of interest and of certain benefit in its own right.