[denial]

Singular integrals are a lovely topic, especially from a Fourier analytic perspective. Taking the Fourier transform, F(H(f))(x) = -i * sgn(x) * F(f)(x), which implies H^2 = - I. H has the Fourier multiplier -i * sgn(x). The Riesz transforms R_j are a higher dimensional generalization of the Hilbert transform with Fourier multipliers -i * x_j/|x_j|, which leads to the nice property sum R_j^2 = -I.

[denial]

I found a small error I can't edit into this comment: the Fourier multiplier of R_j is -i * x_j/|x|.