Comment. Math. Helvetici 6ll (1985) 217-246 0010-2571/85/020217-30501.50 + 0.20/0 1985 Birkh~iuser Verlag, Basel Some weighted norm inequalities concerning the Schr6dinger operators S. Y. A. CHANG, J. M. WmSON and T. H. WOLFF Introduction Let v be a nonnegative, locally integrable function on ~a. Let L =-A-v be the associated Schr6dinger operator. If L is essentially selfadjoint on C~, then positivity of L is equivalent via an integration by parts with IR lUl2vdx<--fR 1~Tul2dx Vu~Co. (0.1) In [8], C. Fefferman asks for conditions on v---0 which imply I~d 'u'2 vdx<-c l Rd IVul2 dx (0.2) for some constant c. By considering translates and dilates of a fixed bump function it is clear that a necessary condition for (0.2) is IO--~l v ax <_ c'/(O) -a (0.3) for some c', for all cubes O cN a. (IQI and/(O) denote the Lebesgue measure and side length of O respectively.) Letting v dx be (approximately) Lebesgue measure on a codimension 2 hyperplane, we see that (0.3) is not sufficient. In [8], it is shown that a sufficient condition is: (~Q~IQ vr' dx) l/P<%l(O) -2 (0.4) Key Words: Schr6dinger operator, weighted norm inequalities, Lusin area function. AMS Subject Classification: 42B25, 47B25, 81C10. 217