Abstract
Let N be a free nilpotent Lie group of step k (GREATERTHEQ) 2 such that the Lie algebra g is stratified and generated by two elements X,Y in V(,1), if g = V(,1)(CRPLUS)(.)(.)(.)(CRPLUS)V(,k). Define L = X + iY. Then we show (')L is hypoelliptic on Lu(VBAR)u is a distribution . That is, if (')LLu(VBAR)(,V )(epsilon) C('(INFIN))(V), then Lu(VBAR)(,V) (epsilon) C('(INFIN))(V). More precisely, if (zeta),(zeta)' (epsilon) C(,O)('(INFIN))(V) such that (zeta)' = 1 in a neighborhood of the support of (zeta), then for each s (epsilon) (//R), s (GREATERTHEQ) 0, there exists C(,s,m) > 0 such that if u (epsilon) H('-m) and (zeta)'(')LLu (epsilon) H('s), then (zeta)Lu (epsilon) H('s+(epsilon)) for some (epsilon) > 0, and (VBAR)(VBAR)(zeta)Lu(VBAR)(VBAR)(,s+(epsilon)) (LESSTHEQ) C(,s,m) ((VBAR)(VBAR)(zeta)'(')LLu(VBAR)(VBAR)(,s) + (VBAR)(VBAR)u(VBAR)(VBAR)(,-m)). Here H('s) denotes the Sobolev space of order s. Also we prove that the operators (')LL + tI and (')LL + tI are hypoelliptic for t (epsilon) (//R)- 0 , where I is the identity map. Let (zeta),(zeta)' be defined as above, then for each s (epsilon) (//R), s (GREATERTHEQ) 0, there exists C(,s) > 0 such that if (zeta)' ((')LL + tI)u (epsilon) H('s), then (zeta)u (epsilon) H('s), and (VBAR)(VBAR)(zeta)u(VBAR)(VBAR)(,s) (LESSTHEQ) C(,s)((VBAR)(VBAR)(zeta)' ((')LL + tI)u(VBAR)(VBAR)(,s) + (VBAR)(VBAR)(zeta)'u(VBAR)(VBAR) (,s-(epsilon)/2)) for some (epsilon) > 0. When k = 2, N is the Heisenberg group ('1) and (')L is the Hans Lewy operator. By using the techniques of microlocal analysis, one can give another proof of the existence of the "relative fundamental solution G" for (')L such that (')L (.) G = I - C(,L) holds for distributions with compact support, where C(,L) is the orthogonal projection onto the nullspace of L. Also some sufficient conditions for solving the equation (')Lu = f globally are given here.