In this talk, we present a necessary and sufficient condition for the \(L^p\) boundedness, for \(p \leq 2\), of parabolic Riesz transforms associated with parabolic operators whose elliptic part is in divergence form with rough coefficients (depending on space and time in a merely measurable way). This yields an extrapolation result from the case \(p = 2\) (the parabolic Kato square root estimate). Our approach relies on new off-diagonal estimates for the parabolic gradient of the resolvent family. These estimates exhibit weak decay, which is nevertheless sufficient for implementing a two-scale Blunck–Kunstmann extrapolation argument, yielding an extrapolation interval that can then be iterated to reach exponents strictly below \(2_\star\). In the case of real coefficients, boundedness holds for all \(p\in(1,2]\), together with a weak-type \((1,1)\) estimate for the spatial gradient component. We shall also discuss the case where the elliptic part is degenerate, in the presence of a Muckenhoupt weight.

 

The talk is based on joint work with Moritz Egert and Benjamin Kosmala (TU Darmstadt, Germany).