UNIQUE CONTINUATION FOR SCHRODINGER OPERATORS WITH POTENTIAL IN MORREY SPACES

¡Au(x)i :5 V(x)Iu(x)I, x E 2, where V is a non smooth, positive potential . We are interested in global unique continuation properties . That means that u must be identically zero on 9 if it vanishes on an open subset of 52 . There is an extensive literature on the matter, mainly to relax the local integrability condition required to the potential V . When Ló, classes are conlo p > n/2 is a necessary and sufficient condition for the strong unique continuation property [JK] (see [K] for references) . In this paper we shall consider some spaces introduced by Morrey [M], which have been recently used by C. Fefferman and D.H. Phong [FP] in studing the eigenvalues of Schrodinger operators ; these spaces contain Lé̀ ~ . We say that V E FP , LP >, with A = 2p n in classical notation [P], if


Introduction
Let us consider in a domain 52 of Rn solutions of the differential inequality ¡Au(x)i :5 V(x)Iu(x)I, x E 2, where V is a non smooth, positive potential.
We are interested in global unique continuation properties.That means that u must be identically zero on 9 if it vanishes on an open subset of 52.
There is an extensive literature on the matter, mainly to relax the local integrability condition required to the potential V. When Ló, classes are conlo p > n/2 is a necessary and sufficient condition for the strong unique continuation property [JK] (see [K] for references) .In this paper we shall consider some spaces introduced by Morrey [M], which have been recently used by C. Fefferman and D.H. Phong [FP] in studing the eigenvalues of Schrodinger operators ; these spaces contain Lé`~.
We say that V E FP , LP >, with A = 2p -n in classical notation [P], if ÍÍVIIFv -SQ (Q12~n -P(IQ where the sup is taken over all cubes in R" and ¡Q1 = Volume of Q .Notice FPCF9ifp>q . In this paper we prove that any solution of (1) has the global unique continuation property if V E Fó, and p > (n -2)/2.Very recently T .Wolf has obtained the same result with a different approach.
We would like to thank C. Kenig for telling us about T. Wolf's result .
The point to obtain this improvement is that in the above works the Carleman estimate is seen as a consequence of a uniform Sobolev inequality (see [KRS]) .
As we shall see while (2) is based on the restriction theorem for the Fourier Transform en the (n-1)-dimensional sphere, together with classical theory of weights, our proof follows from a detailed analysis of the multiplier associated to (3) which just involves the restriction theorem in dimension n-2 .Therefore the assumption in p comes from the restriction operator in the sphere.We think that this is just a technical obstruction and the restriction theorem should be true for p > 1 .Notice that we are close in the case n = 4.We also remark that Fió, contains the so called Kato-Stummel class which B. Simon has conjectured is enough to assure unique continuation (see [S]).
In the sequel we denote by H¡ ,(S2) the classical Sobolev space, and The main theorem is: We define the local Morrey class as the functions W such that IIIWIII = suplimsup IIXB(y,r)(-)W(-)IIF, < oo .yEn r-+0 Theorem 1 .Let u E H¡,(Q), n >_ 3, be a solution of (1), then there exists an e > 0, only depending on p and n, such that if V E Fó, II V II F P < e, p > (n -2)/2.and u vanishes in an open subdomain of 52, then u must be zero everywhere in 52 .
The proof is related to a restriction theorem for the Fourier Transform, obtained in [CS] and [ChR], for which we are going to give an easy proof.Let us define, for this purpose, the Morrey classes ; we say that V is in F',P if IIIVIII«,P = supr°(AV B ( x r )VP) 1 /P < oo, r,x where the sup is taken on all the balls contained in SZ.This notation corresponda to £',P in [P], 1 < a < n/p.Also F2,P = FP .
Theorem 2. Let do, be the uniform measure on the unit sphere Sn -1 in Rn, and (do-)n its Fourier transform, let V E Fa,P , p > (n -1)/2(ce -1), and consider ¡he operator for any f in Có .
Then there exists a constant C such that It would be interesting to understand how this theorem is related to the one in [V] for mixed norm introduced by Rubio de Francia in the study of Bochner-Riesz operators [R] .

. The Carleman estimate
It is standard to obtain Theorem 1 as a consequence of the following Carleman estimate .This reduction can be seen in the case of L 2 weighted estimates in [CS] or [ChR].
On one hand observe that a straightforward calculation gives Imj(0I = I(Kj)^(J)j < Cmin{2j6,1} and, as a consequence, On the other hand for any natural number m there exists a constant Cm such that Consider first the case 0 < j < 1 + [log 1/6] .For k E Z we define Then Finally we can make in Rn a grid with paralellepipeds {Qv} such that the dimension of Q are 2i x ... x 2j x 6-1 .
Call f = f -XQ .Then where Q* is a paralellepiped with the same center as Qv and side ten times bigger than the sides of Q .By (1.4) and Young's inequality Now observe that if w = VPo and V E FP°, then Interpolation with (1.3) gives 1/2 1/z CJ IK' * fl2V) < C611VII FPO (f lf I 2 V -1 ) , if 0 < j < 1 + [log 1/61 .

The Restriction theorem
We give the proof of theorem 2. Let us remak again that this theorem is contained in [CS] and [ChR], but the simplicity of our proof justifies to write it here.
The classical P. Tomas, estimate for the Fourier Transform of Kj(x) gives us the boundedness of Ti = Ki* from L2 to L2 with norm 2j .
It is an open question if the above operator send L2(V -1 ) to L2(V) for V F«,P, p < (n -1)/2 .The answer to this question would be the corner stone extend unique continuation properties to potential in FP for p < (n -2)/2 .