Corrigenda to "unique continuation for Schrödinger operators" and a remark on interpolation of Morrey spaces

Publicacions Matem`atiques, Vol 39 (1995), 405–411. CORRIGENDA TO “UNIQUE CONTINUATION FOR SCHR¨ ODINGER OPERATORS” AND A REMARK ON INTERPOLATION OF MORREY SPACES Alberto Ruiz and Luis Vega Abstract The purpose of this note is twofold. First it is a corrigenda of our paper [RV 1 ]. And secondly we make some remarks concerning the interpolation properties of Morrey spaces. 1. Corrigenda. In our paper “Unique continuation for Schr¨odinger operators with potential in Morrey spaces” [RV1], we claimed the following statement with the name of Theorem 1 —see (5), (6) below for the necessary definitions: “Let u∈H2 loc(Ω),n≥3, be a solution of (1) |∆u(x)|≤|V(x)u(x)|,x∈Ω, and Ωa connected, open subset of Rn. Then there exists an >0, depending just on pand n, such that if V∈Fp loc =L2,p,VL2,p ≤, p > n−2 2, and uvanishes in an open subdomain of Ω, then umust be zero everywhere in Ω”. Unfortunately our proof happens to be incorrect. The theorem is nevertheless true, for T. Wolff obtained a closely related statement by using different arguments, see [W]. Both authors supported in part by Spanish DGICYT grants. 406 A. Ruiz, L. Vega Our approach to unique continuation was based upon the following Carleman estimate: “There exists a constant C>0such that for Vin Fp,p>n−2 2 (2) eτxnuL2(V)≤CVL2,p eτxn∆uL2(V−1), holds for every uin C∞ 0and τin R”. To obtain this inequality we took a global parametrix of the operator eτxn∆e−τxn, which later we realized can not be uniformly bounded in τfor V∈L2,p,p≤(n−1)/2 (one has to multiply the right hand side at least by log τ). In fact, the lemma in page 294 of [RV1] gives the following estimates for a dyadic decomposition Tδof that parametrix : (3) TδfL2(V)≤Cδ|log δ|VL2,p0fL2(V−1),if p0=n−2 2. (4) TδfL2(V)≤Cδ1+VL2,p fL2(V−1),if p>n−2 2. Estimate (3) is true, but (4) holds only for p>(n−1)/2. In fact from (3) and if (n−2)/2≤p≤(n−1)/2 a logarithmic growth of the type Cδ|log δ|is easily obtained. The interesting remark is that this growth turns out to be also necessary and hence, there is no convexity for the bounds of the operator Tδin the range (n−1)/2≤p<(n−2)/2. This fact has some consequences about the interpolation properties in Morrey spaces that we shall consider in Section 2. If we substitute in (2) the Carleman weight τxnby τ(xn+x2 n/2), we can use our approach, as we did in [RV2], to improve the known results on unique continuation of solutions of the inequality (1) when V∈Lα,p,α<2 —see (5), (6) below for the definition. In any case we can not recover Wolff’s result (case α= 2) but only a weaker result for a logarithmic substitute of the space L2,p,p>(n−2)/2. We do not want to get involved in these calculations in the present note. On the other hand we do not know if the inequality (2) is true or false. 2. Interpolation and Morrey-Campanato spaces. Morrey-Campanato classes form a two parameter family of spaces Lα,p,α∈(−1, n/p], p∈[1,∞). We say that f∈L α,p,iffis in Lp loc and there exists a constant C>0, which depends on f, such that for every x∈Rnand every r>0, we can find a number σ∈R, which depends on f,x, and rsuch that (3) rαr−nQ(x,r) |f(y)−σ|pdy1/p <C, Corrigenda and interpolation 407 where Q(x, r) is a cube centered at xand volume rn. The case α>0 was introduced by Morrey in the study of the regularity problem of the Calculus of Variations. It was proved by Campanato that σcan be taken zero without loss of generality —see [C1]. Then for α>0 and f∈L α,p we define (4) fLα,p = sup x sup r rαr−nQ(x,r) |f(y)|pdy1/p . Notice that if α>0 and p=n/α,p≥1 we obtain the Lebesgue space Lp. The class was extended by Campanato to α≤0 and he [C2] and Meyers [M] proved to be the space of (−α)-Holder continuous functions; this theorem is known as the integral characterization of Holder-continuous functions. When α= 0 we have John-Niremberg space BMO. Notice that in this case, i.e. α≤0, the space does not change with pif αis fixed. Recently Morrey classes, i.e. α>0, have been the object of some works on weighted Sobolev estimates, unique continuation properties —see [FP], [CS], [ChR], [W], [RV2], and some other problems in PDE —see [RV3], and [T] for example. In this note we are concerned with interpolation properties of Morrey spaces for α>0. In particular we prove the lack of the convexity which characterizes interpolation functors of exponent θ—see [BL, p. 27]. In particular the complex and real methods have this property. The interpolation properties of Morrey-Campanato spaces have been studied in several works during the 60’s —see [S], [P] and the references there in. In particular, Stampacchia, [S], and Campanato and Murthy, [CM] proved that for Ta linear operator, and TLqi→Lαi,pi=Ki, with 1 ≤pi,qi≤∞,αi>0, i=1,2, then Tis bounded from Lqθ to Lαθ,pθwith norm at most KK1−θ 1Kθ 2, where 1 pθ=(1−θ)1 p1+θ1 p2, 1 qθ=(1−θ)1 q1+θ1 q2, and αθ=(1−θ)α1+θα2, and Kdepending just on θ,αi,pi,qi,i=1,2. In the more general setting of Morrey-Campanato classes, Stein and Zygmund, [StZ], constructed a linear operator bounded from Lα,p to Lα,p for some α<0 and from L2to L2, which is not bounded from BMO to BMO. Let us remark that BMO = L0,p and L2=Ln/2,2. Hence interpolation through the line α= 0 does not hold. We have the following result. 408 A. Ruiz, L. Vega Theorem. Set n>1and 0<α<n. Then, given any p1,p2,p3, and C>0, such that 1≤p2<p 3<n−1 α<p 1<∞, and C>0, there exists a continuous linear operator T:Lα,pi→L1,i=1,2,3, such that TLα,pi→L1≤Ki, i=1,2and (4) TLα,p3→L1>CK 1−θ 1Kθ 2, for 1 p3=(1−θ)1 p1+θ1 p2. Proof of the theorem: Let φbeaC∞ 0non negative function such that φ≤1, φ(x)=1 if |x|<1/4, and φ(x)=0if|x|>1/2. For 0 <δ<1 consider the operator Tgiven by multiplication by Φδ(x)=φ(|x|)φ(δxn), with x=(x1,... ,x n−1): Tf(x)=Φ δ(x)f(x). Then, |Tf|=Φδ|f| ≤δ−(1−1/p)|Φδf|p1/p ≤δ−(1−1/p)δα−n/pfLα,p . And also |Tf|= νQν Φδ|f|, where Qνis a collection of δ−1cubes of volume one. Hence, |Tf|≤δ−1fLα,p. Therefore TLα,p→L1≤δ−1+α−(n−1)/p,if p≥(n−1)/α δ−1,if 1 ≤p≤(n−1)/α. Now take f=Φ δ(x). Then there exists a dimensional constant cn such that |Tf|≥cnδ−1, Corrigenda and interpolation 409 and fLα,p =c−1 nδ−α+(n−1)/p,if p≥(n−1)/α c−1 n,if 1 ≤p≤(n−1)/α. Therefore TLα,p→L1≥c2 nδ−1,if 1 ≤p≤(n−1)/α. Fix α>0 and take p1and p2such that (n−1)/α ≤p1≤n/α, and 1≤p2≤(n−1)/α. Then, on one hand, for θ∈(0,1), we have Tθ Lα,p1→L1T1−θ Lα,p2→L1≤δ−1+, where =(α−(n−1)/p1)θ>0. On the other hand for p2<p 3<(n−1)/α,wehave TLα,p3→L1≥c2 nδ−1. Taking δsmall enough we have proved the theorem. Final remarks. The above theorem can be extended to a more general situation. In particular the restriction on the dimension, α, and pscan be avoided. In fact we have an example of a bounded linear operator which is unbounded in a given intermediate space. Therefore Morrey spaces are not closed by interpolation in a strong form, and not just by the lack of convexity. Writing this example would have made this note too large and getting us too far from the initial purpose which is the corrigenda of our previous paper. On the other hand the counterexample given in the theorem, is a small variation of the one needed for the inequality (4). We have preferred to write it in this way, to illustrate that the lack of convexity is due to the particular structure of Morrey spaces and their bad behaviour with respect to interpolation. Details about the above questions will appear elsewhere. Finally we would like to thank J. Peetre for sharing with us his belief that Morrey spaces are not closed under interpolation. 410 A. Ruiz, L. Vega References [BL] Bergh, J. and Lofstrom, J.,“Interpolation Spaces,” SpringerVerlag, New York, 1976. [C1] Campanato, S., Proprieta di una famiglia di spazi funzionali, Ann. Scuola N. Sup. Pisa 18 (1964), 137–160. [C2] Campanato, S, Proprieta di h¨olderianita di alcune classi de funzioni, Ann. Scuola N. Sup. Pisa 17 (1963), 175–188. [CM] Campanato, S. and Murthy, M. K. V., Una generalizzazione del teoremi de Riesz–Thorin, Ann. Scuola N. Sup. Pisa 19 (1965), 87–100. [CS] Chanillo, S. and Sawyer, E., Unique continuation for ∆+ V and the C. Fefferman-Phong class, Trans. AMS 318(1) (1990), 275–300. [ChR] Chiarenza, F. and Ruiz, A., Uniform L2-weighted Sobolev inequalities, Proc. AMS 112(1) (1991), 53–64. [FP] Fefferman, C. and Phong, D. H., Lower bounds for Schrodinger equations, J. Eq. aux Derivees Partielles, Saint Jean de Monts, Soc. Mat. de France (1982). [M] Meyers, G. N., Mean oscillation over cubes and Holder continuity, Proc. AMS. 15 (1964), 717–721. [P] Peetre, J., On the theory of Lp,λ spaces, Journal of Functional Analysis 4(1969), 71–87. [RV1] Ruiz, A. and Vega, L., Unique continuation for Schrodinger operators in Morrey spaces, Publicacions Matematiques 35 (1991), 291–298. [RV2] Ruiz, A. and Vega, L., Unique continuation for the solutions of the Laplacian plus a drift, Ann. Ins. Fourier, Grenoble 41(3) (1991), 651–663. [RV3] Ruiz, A. and Vega, L., Local regularity of solutions to wave equations with time-dependent potentials, Duke Math. J. 76(3) (1994), 913–940. [S] Stampacchia, G.,L(p,λ)- Spaces and interpolation, Comm. in Pure and App. Math. 17 (1964), 293–306. [StZ] Stein, E. M. and Zygmund, A., Boundedness of translation invariant operators on Holder spaces and Lp-spaces, Ann. Math. 85 (1967), 337–349. [T] Taylor, M., Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. in PDE. 17 (1992), 1407–1456. Corrigenda and interpolation 411 [W] Wolff, T., Unique continuation for |∆u|≤V|∇u|and related problems, Revista Matem´atica Iberoamericana 6(3) (1990), 155–200. 1991 Mathematics subject classifications: 42B20, 42B25 Alberto Ruiz: Departamento de Matem´aticas Universidad Aut´onoma de Madrid 28049 Madrid SPAIN e-mail: [email protected] Luis Vega: Departamento de Matem´aticas Universidad del Pa´ıs Vasco Apartado 644 48080 Bilbao SPAIN e-mail: mtpv[email protected]u.es Primera versi´o rebuda el 19 de Gener de 1995, darrera versi´o rebuda el 19 de Setembre de 1995