# Riesz transform of fractional operators

I am interested in Riesz transforms linked to the fractional Laplacian and other fractional operators. I have been hunting down in the literature to find related results but I have not been able to find any.

Any related references would be greatly appreciated.

• I do not think it is immediately clear what is the notion of a Riesz transform corresponding to a fractional operator. Do you mean something like $\partial_x L^{-1}$, where, say, $L = (-\Delta)^s$? – Mateusz Kwaśnicki May 24 '20 at 8:53
• I am looking for Riesz transforms of the form $D L^{-1/2}$ with $L=(-\Delta)^{\alpha/2}$ $\alpha \in (0,2)$ and $D$ some (fractional) derivative operator. – user69642 May 24 '20 at 10:16
• Then it looks like you are after fractional powers of the usual Riesz transforms, right? I do not think these have been studied much, but general theory of singular integrals should apply. (By the way, out of curiosity: what is your motivation to study these operators?) – Mateusz Kwaśnicki May 24 '20 at 21:24
• Meyer's inequality with a stable reference measure – user69642 May 24 '20 at 21:39
• Some results on the Riesz transform for the fractional Laplacian with (a kind of) fractional derivative are contained in this paper by Junge, Mei and Parcet: arxiv.org/pdf/1407.2475.pdf (see 1.4.A). Is this something of the kind you were looking for? – MaoWao Jan 27 at 16:37

You can find an elementary and very detailed approach to fractional operators and Riesz transforms in

See in particular the last section, pages 109-117. I hope it will be helpful.

• Thank you for the reference and your answer. I am looking for something intrinsically fractional for the "first order" derivative operator used to define the Riesz transform. – user69642 May 24 '20 at 21:05