Discrete analogues in harmonic analysis: $T{T}^{\ast }$ methods

In this note we present how the almost-orthogonality methods based on $T{T}^{\ast }$ arguments can be employed to study boundedness of discrete operators of Radon type. Almost-orthogonality methods have particular significance when the classical Fourier methods are not available. However here, to avoid technicalities and present the key ideas behind the discrete almost-orthogonality methods, we give a new proof of the ${\mathrm{\ell }}^2({\mathbb{Z}}^d)$-boundedness of Bourgain’s maximal inequality for Radon polynomial averages.