In the paper "A Convolution Inequality Concerning Cantor-Lebesgue Measures", the Littlewood-Paley theory is used to estimate the norm of multiplier operator in Lemma 1. It is claimed that Lemma 2 is an elementary variant and the proof involves the boundedness of Hilbert transform, but the proof is not given.

- How can we prove Lemma 2? How is it related to Hilbert transform?
- Is there any version of Littlewood-Paley theorem for decomposition into finitely many pieces? Will this help proving Lemma 2?
- It is claimed that he constants involved in Lemma 1 and 2 will tend to $1$ as $p,q$ tend to $2$. Why is this true? Do these follow from Riesz-Thorin interpolation theorem?

I will be grateful for any help and hints~