A priori estimates related to differential operators of Kuptsov--Hörmander type (Q1778294)

This paper deals with a-priori estimate of the type \[ \| a(D)u\|_s\leq C_1\| Pu\|_s+ C_2\| u\|_0,\quad s> 0,\quad\forall u\in C^\infty_0(K),\tag{1} \] where the operator \(P(x,D)\) is a sum of squares of real valued \(C^\infty\) smooth vector fields \(A_j\) and the symbol \(a(\xi)\to\infty\) for \(|\xi|\to \infty\). The author assumes that at each point \(x\) at least one of the vector fields is nonzero, two arbitrary close points \(x\), \(y\) can be joined by arcs of integral curves of the vector fields \(A_j\) or their linear combinations and the sum of lengths of these arcs does not exceed \(\rho(z)\), \(\rho(0)= 0\), \(z= (x-y)\). Then he proves the validity of (1) for \(a=\rho^{-2}(1+ |\xi|^2)^{-1/2}\). Moreover, the operator \(P(x,D)\) is hypoelliptic if \(\lim_{\sigma\to 0}\ln(\sigma)\rho(\sigma)= 0\).