Riesz transforms on Euclidean space

Assume Countable Choice, let $n\ge 1$, and let $1\le j\le n$. On $L^2({\mathbb{R}}^n;\mathbb{C})$ define the $j$-th Riesz transform $R_j$ by its Fourier multiplier

$m_j(\xi ):=\{\begin{array}{ll}-i{\xi }_j\mathrm{/}\mathrm{\mid }\xi \mathrm{\mid },& \xi \mathrm{\ne }0,\\ 0,& \xi =0,\end{array}\widehat{R_{j}f}=m_j\widehat{f},$

where the Fourier transform is the unitary Plancherel extension ${\mathcal{F}}_2$ of Plancherel theorem and the multiplier acts by $R_j={\mathcal{F}}_2^{-1}{M}_{m_j}{\mathcal{F}}_2$. The symbol $m_j$ is measurable and $\mathrm{\mid }{m}_j(\xi )\mathrm{\mid }\le 1$ for every $\xi$ on account of $\mathrm{\mid }{\xi }_j\mathrm{\mid }\le \mathrm{\mid }\xi \mathrm{\mid }$, so the published $L^2$ multiplier theorem Exact L2 Fourier multiplier norm applies with essential supremum at most one: $R_j$ is a well-defined bounded complex-linear operator on $L^2({\mathbb{R}}^n)$, it depends only on the almost-everywhere class of $m_j$, and in particular the assigned value $m_j(0)=0$ has no effect on the operator. The theorem also identifies $R_j$ on Schwartz functions with the regular distribution of ${\mathcal{F}}_2^{-1}(m_j{\mathcal{F}}_{2}f)$, and gives $\mathrm{\parallel }{R}_j\mathrm{\parallel }\le 1$.

The Riesz kernel attached to this definition is the function on ${\mathbb{R}}^n\setminus \{0\}$

$K_j(x):=c_n\frac{x_j}{\mathrm{\mid }x{\mathrm{\mid }}^{n+1}},c_n:=\frac{\mathrm{\Gamma }((n+1)\mathrm{/}2)}{{\pi }^{(n+1)\mathrm{/}2}},$

with $\mathrm{\Gamma }$ the Euler integral. Since $(n+1)\mathrm{/}2>0$, the published convergence theorem Euler's Gamma integral converges exactly for positive real parameters gives $0<\mathrm{\Gamma }((n+1)\mathrm{/}2)<\mathrm{\infty }$, so $c_n$ is a positive finite constant and $K_j$ is a smooth function on ${\mathbb{R}}^n\setminus \{0\}$, odd under $x\mapsto -x$ and homogeneous of degree $-n$: $K_j(tx)=t^{-n}{K}_j(x)$ for $t>0$.

This definition asserts only the multiplier description. It does not assert that the principal value ${\lim }_{\epsilon \downarrow 0}{\int }_{\mathrm{\mid }y\mathrm{\mid }>\epsilon }{K}_j(y)f(x-y)dy$ exists for any particular $f$ or $x$; that statement is proved separately for Schwartz functions, as is the identification of the limit with the $L^2$ class $R_{j}f$. In dimension $n=1$ the constant collapses to $c_1=\mathrm{\Gamma }(1)\mathrm{/}\pi =1\mathrm{/}\pi$, by the value $\mathrm{\Gamma }(1)=1$ of The real Gamma functional equation $\mathrm{\Gamma }(s+1)=s\mathrm{\Gamma }(s)$, and $K_1(x)=1\mathrm{/}(\pi x)$ is the line Hilbert kernel; the comparison of $R_1$ with the Hilbert transform of the line is worked out on the examples page. The Fourier convention is the $e^{-2\pi ix\cdot \xi }$ convention of ${\mathcal{F}}_2$. Replacing its phase by $e^{-ix\cdot \xi }$ leaves both $m_j$ and $c_n$ unchanged: the frequency rescaling $\xi \mapsto \xi \mathrm{/}(2\pi )$ preserves ${\xi }_j\mathrm{/}\mathrm{\mid }\xi \mathrm{\mid }$.