An application of homogenization theory to harmonic analysis on solvable Lie groups of polynomial growth (Q1314983)

Let \(Q\) be a connected simply connected solvable Lie group of polynomial growth. Let also \(E_ 1,\dots, E_ p\) be left invariant vector fields on \(Q\) that satisfy Hörmander's condition and denote by \(L=- (E_ 1^ 2+ \cdots+ E_ p^ 2)\) the associated sub-Laplacian. The author proves the following Harnack inequality (*): there exists a constant \(c\geq 0\) such that \(| E_ i u(x)|\leq ct^{-1} u(x)\), \(x\in Q\), \(t\geq 1\), \(1\leq i\leq p\), for all \(u\geq 0\) such that \(Lu=0\) in the ball \(S(x,t)\) centered at \(x\) of radius \(t\) for the control distance associated to the vector fields \(E_ i\). In the proof the author introduces a limit group \(Q_ H\), which is a nilpotent homogeneous Lie group and which via a special diffeomorphism \(\varphi\) has the same manifold structure as \(Q\). Furthermore there exists a homogeneous limit operator \(L_ 0\) on \(Q_ H\), which is a sum of squares of invariant vector fields. The dilations \(\tau(\varepsilon)\), \(\varepsilon \geq 0\) of \(Q_ H\), define also via \(\varphi\) ``dilations'' of \(Q\) and hence for every \(\varepsilon>0\) we have an operator \(L_ \varepsilon\) on \(Q\), such that \(Lu=0\) is equivalent with \(L_ \varepsilon u_ \varepsilon=0\), where \(u_ \varepsilon(x)= u(\tau_ \varepsilon (x))\), \(x\in Q\). We have that \(\lim_{\varepsilon\to 0} L_ \varepsilon= L_ 0\). The author shows that there exists \(\varepsilon_ 0>0\), \(c>0\) such that for all \(\varepsilon\in (0,\varepsilon_ 0]\) and for all \(u_ \varepsilon\) with infinity norm \(\leq 1\) living on the unit ball and satisfying \(L_ \varepsilon u_ \varepsilon=0\), we have that \[ \sup_{x\in D_{\varepsilon/\varepsilon_ 0}} | u_ \varepsilon(x)- A_ 0^ \varepsilon|< c\varepsilon/ \varepsilon_ 0 \] where \(A_ 0^ \varepsilon\) is a constant such that \(A_ 0^ \varepsilon <c\). This result combined with a Moser type Harnack inequality of N. Th. Varopoulos gives the proof of the Harnack inequality (*). Finally the author shows by an example of a group of polynomial growth which is not nilpotent that (*) is in some sense optimal.