Why the Riesz transforms are averages of the dyadic shifts? (Q1861100)

Let \(\mathcal{L}(k)\) be the dyadic grid of intervals with the length \(2^{-k}, k\in \mathbb{Z}\), and let \(\mathcal{D}_t\) be grid of the type \(\mathcal{L}(0)\) such that \(t\) is one of the end-points of its intervals. The authors consider the operator \[ \mathbb{P}_tf=\sum_{I\in \mathcal{D}_t}(f,h_I)\chi_I(x) \] where \(h_I(x)=-1\) on the left half of \(I\) and \(h_I(x)=1\) on its right half. The authors average this operator. Since the average is invariant with respect to translations, it is a convolution. They show that the kernel of this convolution is nothing else but \(\frac{1}{4x}\), so that this average reduces to the Hilbert transform. In the case of two-dimensional dyadic lattices, they similarly arrive at a singular operator with the kernel related to the Riesz transform (of the type \(\frac{x_1}{|x|^3}\)). Some geometric applications are given.