7 problems

Rockland's conjecture. The Rockland condition is equivalent to hypoellipticity:

Let $P(D)$ be a hypoelliptic partial differential operator on $\mathbb{R}^d$ which is orthogonally degenerate on its linear characteristic cone … Moreover, let $F_1\subset F_2$ be…

Hypoelliptic damping conjecture. The second part of the theorem for the elliptically perturbed operator $P_\epsilon$ holds with $\widetilde P_\epsilon$ in place of $P_\epsilon$. Th…

Let $P(\Delta)$ be an operator on a product of p.c.f. fractals, and let $P_0$ be the principal part of $P$. The operator is quasielliptic if there is a constant $c>0$ such that … f…

Let $P$ be a polynomial in $m$ variables and let $P(\Delta)$ be the corresponding operator on a product of p.c.f. fractals. For a multi-index $\kappa$, write $D^{\kappa}P$ for the…

Let $X$ be a p.c.f. fractal, let $P$ be a polynomial, and let $P(Delta)$ denote the associated operator on the product space $X^m$. The operator $P(Delta)$ is elliptic when…

The enthralled-vector-fields conjecture. The real vector fields $X$ satisfying this estimate coincide on a dense open subset of $M$ with the elements of the…