SAK principle for a class of Grushin-type operators (Q854551)

In his famous paper [Bull. Am. Math. Soc., New Ser. 9, 129--206 (1983; Zbl 0526.35080)], \textit{C. Fefferman}, building on joint work with \textit{D. H. Phong}, suggests a strategy to get a-priori estimates for pseudodifferential operators, the so-called SAK-principle. Given second order symbols \(p,q\in S^2({\mathbb R}^n)\) (the standard class of pseudodifferential operators on \({\mathbb R}^n\)), with \(p\geq 0,\) the problem is to find conditions on \(q\) in terms of the geometry of \(p\) in order to get the inequality: there exists \(C>0\) such that \[ \|q^w(x,D)u\|_0\leq C\Bigl(\|p^w(x,D)u\|_0+\text{ ``small error''}\Bigr),\quad\forall u\in C_0^\infty.\tag{1} \] Here \(q^w(x,D)\) denotes the Weyl-quantization of the symbol \(q\), and \(\|u\|_0\) is the \(L^2\) norm of \(u\). This is in analogy with the Fefferman-Phong inequality, which says that if \(0\leq p\in S^2\) then \[ (p^w(x,D)u,u)\geq -C\|u\|_0^2,\quad \forall u\in C_0^\infty\leqno(2) \] (see also the paper of \textit{N. Lerner} and \textit{J. Nourrigat} [Ann. Inst. Fourier 40, No. 3, 657--682 (1990; Zbl 0703.35182)], in which they show in the \(1\)-dimensional case that in case the set where the symbol \(p\) is \(\leq 0\) is ``symplectically small'', then estimate \((2)\) holds). Fefferman conjectured that \((1)\) holds when \[ \max_{B_\nu}|q(x,\xi)|\leq c\max_{B_\nu}p(x,\xi), \] where \(\{B_\nu\}_{\nu\in{\mathbb Z}_+}\) is a suitable ``partition'' of the phase space \({\mathbb R}^n\times{\mathbb R}^n\) associated with \(p\) (see [\textit{A. Parmeggiani}, Adv. Math. 131, No. 2, 357--452 (1997; Zbl 0940.35214)], where a direct definition is proposed for the candidate balls \(B_\mu\)). \textit{F. Hérau} proved in [Ann. Inst. Fourier 50, No. 4, 1229--1264 (2000; Zbl 0956.35141)] that \((1)\) holds when \(n=1\) assuming that \[ |q(x,\xi)|\leq cp(x,\xi),\quad \forall (x,\xi)\in{\mathbb R}\times{\mathbb R}. \] In this very nicely written and deep paper, the authors prove \((1)\) in the case \(n=2\) and \(p^w(x,D)\) is the Grushin operator in \({\mathbb R}^2\), whose symbol is \(p(x,\xi)=\xi_1^2+x_1^{2h}\xi_2^2\), \(h\in{\mathbb Z}_+\). (Similar arguments work in higher dimension for polynomial symbols having the same kind of anisotropic vanishing behavior.) More precisely, after introducing a suitable Hörmander-metric \(G\) on \({\mathbb R}^2\times{\mathbb R}^2\) tailored to the symbol \(p\), the authors prove the following theorem. Theorem. Let \(q\in S^2\) and suppose there exists \(r\in (0,c_0],\) where \(c_0\) is the slowness constant of \(G\), and a covering of \({\mathbb R}^2\times{\mathbb R}^2\) made of \(G\)-balls \(B^G_{x_\nu,\xi_\nu}(r)\) centered at \((x_\nu,\xi_\nu),\) \(\nu\in{\mathbb Z}_+\), of radius \(r\), such that \[ \max_{B^G_{x_\nu,\xi_\nu}(r)}|q(x,\xi)|\leq c\max_{B^G_{x_\nu,\xi_\nu}(r)}p(x,\xi),\quad \forall \nu\in{\mathbb Z}_+, \] where \(c>0\) is independent of \(\nu.\) Then for any given compact \(K\subset{\mathbb R}^2\) there exists \(C_K>0\) such that \[ \|q^w(x,D)u\|_0\leq C\Bigl(\|p^w(x,D)u\|_0+\|u\|_0\Bigr), \quad \forall u\in C_0^\infty(K). \] In Section 2 of the paper, the authors introduce the two levels of microlocalization that are crucial for the proof of the theorem, which is given in Section 3. Finally, a series of technical results used in the proof are given in the Appendix.