Sharp and optimal inequalities in harmonic analysis

This thesis deals with three topics in Harmonic Analysis: 1. Sharp restriction theory; 2. Sparse domination for square function operators; 3. Two weight theory for the Bergman projection. In the first part we study some sharp inequalities that arise composing a \(k\)-plane transform with the square of the Fourier extension operator from the paraboloid. We study the sharp form of these inequalities. We compute the optimal constants and characterise maximisers. The second and main part of this thesis develops on sparse domination for square function operators. In particular we derive a sparse domination in form under minimal testing conditions. We called this domination a “quadratic” as it dominates the non-linear operator \( (Sf)^2\) rather than \(Sf\). This produces optimal weighted estimates for the dominated square functions. We show that a quadratic domination holds also for non-integral square functions associated with a general elliptic operator \(L\). This refines and improves the domination in [BFP16] when the operator is a square function. The last part of the thesis studies the Bergman projection \(P\) on the complex unit ball \(\mathbb{B}^d\) in \(\mathbb{C}^d\). We derive sufficient conditions for two weight estimates for \(P\) via sparse domination. These conditions are given in terms of “bumped” Orlicz averages of the two weights. On the way, we also derive mixed \(B_2\)–\(B_\infty\) estimates for the Bergman projection on \(L^2(\mathbb{B}^d)\)