Riesz transform

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 R"d" are defined by
for "j" = 1,2...,"d". The constant "c""d" is a dimensional normalization given by
where ω"d"−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
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 "L"2(R"d") to itself.
This homogeneity property can also be stated more directly without the aid of the Fourier transform. If σ"s" is the dilation on R"d" by the scalar "s", that is σ"s""x" = "sx", then σ"s" defines an action on functions via pullback:
The Riesz transforms commute with σ"s":
Similarly, the Riesz transforms commute with translations. Let τ"a" be the translation on R"d" along the vector "a"; that is, τ"a"("x") = "x" + "a". Then
For the final property, it is convenient to regard the Riesz transforms as a single vectorial entity "R"ƒ = ("R"1ƒ...,"R""d"ƒ). Consider a rotation ρ in R"d". The rotation acts on spatial variables, and thus on functions via pullback. But it also can act on the spatial vector "R"ƒ. The final transformation property asserts that the Riesz transform is equivariant with respect to these two actions; that is,
These three properties in fact characterize the Riesz transform in the following sense. Let "T"=("T""1"...,"T""d") be a "d"-tuple of bounded linear operators from "L"2(R"d") to "L"2(R"d") such that
Then, for some constant "c", "T" = "cR".
Relationship with the Laplacian.
Somewhat imprecisely, the Riesz transforms of formula_7 give the first partial derivatives of a solution of the equation
where Δ is the Laplacian. Thus the Riesz transform of formula_7 can be written as:
In particular, one should also have
so that the Riesz transforms give a way of recovering information about the entire Hessian of a function from knowledge of only its Laplacian.
This is now made more precise. Suppose that formula_12 is a Schwartz function. Then indeed by the explicit form of the Fourier multiplier, one has
The identity is not generally true in the sense of distributions. For instance, if "formula_12" is a tempered distribution such that formula_15, then one can only conclude that
for some polynomial formula_17.