Riesz transform

Summary

The Riesz transforms $R_1, \ldots, R_n$ are the $n$ singular integral operators on $\R^n$ with Fourier multipliers $-i\xi_j/\abs{\xi}$. They are the higher-dimensional analogue of the Hilbert transform, which is the case $n = 1$. They express second derivatives through the Laplacian, by $\partial_j\partial_k = -R_j R_k \Delta$, so their $L^p$ bounds give $L^p$ estimates for elliptic equations. They are bounded on $L^p(\R^n)$ for $1 < p < \infty$ and commute with translations and dilations.