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 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 The Riesz transforms arises in the study of differentiability properties of harmonic potentials in potential theory and harmonic analysis.
A particular consequence of this last observation is that the Riesz transform defines a bounded linear operator from L2(Rd) to itself.
[1] This homogeneity property can also be stated more directly without the aid of the Fourier transform.
Then For the final property, it is convenient to regard the Riesz transforms as a single vectorial entity Rƒ = (R1ƒ,...,Rdƒ).
Let T=(T1,...,Td) be a d-tuple of bounded linear operators from L2(Rd) to L2(Rd) such that Then, for some constant c, T = cR.
give the first partial derivatives of a solution of the equation where Δ is the Laplacian.
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.
Then indeed by the explicit form of the Fourier multiplier, one has The identity is not generally true in the sense of distributions.