Sharp Riesz and reverse Riesz inequalities for degenerate elliptic operators of Grushin type

Abstract

We determine the sharp $L^p$ ranges of the Riesz transform and the reverse Riesz inequality for a class of power-degenerate elliptic operators of Grushin type. When the degenerate variable has dimension at least two, the Riesz transform is bounded throughout the reflexive range. In dimension one, a capacity-driven transition occurs: weak degeneracy produces a sharp finite upper endpoint, whereas strong degeneracy separates the two half-lines and restores the full range. The reverse inequality displays a different sharp phenomenon. In the weakly degenerate one-dimensional regime it holds throughout the reflexive range except at a single Hardy-critical exponent, while in all remaining regimes it holds for every $1<p<\infty$.

arXiv comment78pages