Riesz transform - Knowlpedia

On $L^2(\mathbb R^n)$ , the Riesz transform $R_j$ , $1\le j\le n$ , is the Fourier multiplier

\widehat{R_jf}(\xi)=\frac{i\xi_j}{|\xi|}\widehat f(\xi),\qquad \xi\ne0.

The symbol may be assigned any value at zero, a null set. This entry uses the plus- $i$ convention; the alternative minus- $i$ convention reverses each individual transform.

Products and integrability

The double transform has symbol $-\xi_i\xi_j/|\xi|^2$ , unchanged if the sign convention for all single transforms is reversed. Riesz transforms extend boundedly to $L^p$ for $1<p<\infty$ , by singular-integral theory. This does not give a bounded map $L^1\to L^1$ . A bounded frequency formula for a specific $L^1$ input can instead be interpreted as a negative-order Sobolev distribution.

References

Tao, Fourier analysis lecture notes 4, §§2 and 4.