Mikhlin–Hörmander multiplier conjecture for the Grushin operator

Let $G$ be the Grushin operator on the relevant Euclidean space, and let $m:\mathds{R}^+\to\mathds{C}$ be a bounded Borel function satisfying the multiplier condition

$$\sup_{t\in [1,\infty[}\|\eta(\cdot)m(t\cdot)\|_{H_s}<\infty.$$

Mikhlin–Hörmander multiplier conjecture. The operator $m(G)$ is bounded on $L_p$ for $1<p<\infty$ and is of weak type $(1,1)$ whenever $s>1$.

This conjecture proposes the analogue for the Grushin operator of the optimal multiplier theorems known for the Heisenberg group and $SU(2)$, with the endpoint excluded.