A counterexample to the \(L^ 2\) estimate \(\| XYu \|\leq C(\| X^ k u\|+ \| Y^ k u\|+ \| u\|)\) (Q1906506)

An example of two \(C^\infty\) vector fields \(X\), \(Y\) on \(\mathbb{R}^2\) is constructed with the property that for any natural number \(k\) none of the following estimates holds. \[ |XY u|\leq C(|X^k u|+ |Y^k u|+ |u|),\;|(XY+ YX) u|\leq C(|X^k u|+ |Y^k u|+ |u|), \] \[ |[X, Y] u|\leq C(|X^k u|+ |Y^k u|+|u|). \] It is shown that the above estimates fail not only in \(L^2\), but in any \(L^p\), \(1\leq p\leq \infty\) and cannot be saved by adding terms of the form \(|Y^k X^\ell u|\) to the right-hand side. Finally, in the given example the operator \(X^2+ Y^2\) is hypoelliptic. A small modification of the example produces two vector fields in \(\mathbb{R}^3\) generating a pseudoconvex CR-structure.