In the mathematical theory of harmonic analysis, the Riesz transforms are a family of generalizations of the Hilbert transform to Euclidean spaces of dimension d > 1. They are a type of singular integral operator, meaning that they are given by a convolution of one function with another function having a singularity at the origin. Specifically, the Riesz transforms of a complex-valued function ƒ on Rd are defined by
for j = 1,2,...,d. The constant cd is a dimensional normalization given by
c d = 1 π ω d − 1 = Γ [ ( d + 1 ) / 2 ] π ( d + 1 ) / 2 . {\displaystyle c_{d}={\frac {1}{\pi \omega _{d-1}}}={\frac {\Gamma [(d+1)/2]}{\pi ^{(d+1)/2}}}.}
where ωd−1 is the volume of the unit (d − 1)-ball. The limit is written in various ways, often as a principal value, or as a convolution with the tempered distribution
K ( x ) = 1 π ω d − 1 p . v . x j | x | d + 1 . {\displaystyle K(x)={\frac {1}{\pi \omega _{d-1}}}\,p.v.{\frac {x_{j}}{|x|^{d+1}}}.}
The Riesz transforms arises in the study of differentiability properties of harmonic potentials in potential theory and harmonic analysis. In particular, they arise in the proof of the Calderón-Zygmund inequality (Gilbarg & Trudinger 1983, §9.4).
Multiplier properties The Riesz transforms are given by a Fourier multiplier. Indeed, the Fourier transform of Rjƒ is given by
F ( R j f ) ( x ) = − i x j | x | ( F f ) ( x ) . {\displaystyle {\mathcal {F}}(R_{j}f)(x)=-i{\frac {x_{j}}{|x|}}({\mathcal {F}}f)(x).}
In this form, the Riesz transforms are seen to be generalizations of the Hilbert transform. The kernel is a distribution which is homogeneous of degree zero. A particular consequence of this last observation is that the Riesz transform defines a bounded linear operator from L2(Rd) to itself. This homogeneity property can also be stated more directly without the aid of the Fourier transform. If σs is the dilation on Rd by the scalar s, that is σsx = sx, then σs defines an action on functions via pullback:
σ s ∗ f = f ∘ σ s . {\displaystyle \sigma _{s}^{*}f=f\circ \sigma _{s}.}
The Riesz transforms commute with σs:
σ s ∗ ( R j f ) = R j ( σ x ∗ f ) . {\displaystyle \sigma _{s}^{*}(R_{j}f)=R_{j}(\sigma _{x}^{*}f).}
Similarly, the Riesz transforms commute with translations. Let τa be the translation on Rd along the vector a; that is, τa(x) = x + a. Then
τ a ∗ ( R j f ) = R j ( τ a ∗ f ) . {\displaystyle \tau _{a}^{*}(R_{j}f)=R_{j}(\tau _{a}^{*}f).}
For the final property, it is convenient to regard the Riesz transforms as a single vectorial entity Rƒ = (R1ƒ,...,Rdƒ). Consider a rotation ρ in Rd. The rotation acts on spatial variables, and thus on functions via pullback. But it also can act on the spatial vector Rƒ. The final transformation property asserts that the Riesz transform is equivariant with respect to these two actions; that is,
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