4 problems

Let $V\in\mathcal K_0$ and let $H=-\Delta+V$. For the intertwining operators involving powers of $H$ and $-\Delta$, consider their boundedness on $L^p$. Optimal-range conjecture. A…

Let $\mathbb{G}$ be a graded Lie group with $\dim(\mathbb{G})\geq 2$, let $\mathcal{G}$ be a positive Rockland operator on $\mathbb{G}$, and let $\chi$ be the characteristic functi…

Dimension-reduction conjecture. The previous theorems should remain true with $D=d_1+d_2$ in place of $D=\max\{d_1+d_2,2d_2\}$.

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 $s>1$.