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

Publicacions Ma em`a iques, Vol 39 (1995), 405–411. CORRIGENDA TO “UNIQUE CONTINUATION FOR SCHR¨ ODINGER OPERATORS” AND A REMARK ON INTERPOLATION OF MORREY SPACES Albe o Ruiz and Luis Vega Abs ac The pu pose o his no e is wo old. Fi s i is a co igenda o ou pape [RV 1 ]. And secondly we make some ema ks conce ning he in e pola ion p ope ies o Mo ey spaces. 1. Co igenda. In ou pape “Unique con inua ion o Sch ¨odinge ope a o s wi h po- en ial in Mo ey spaces” [RV1], we claimed he ollowing s a emen wi h he name o Theo em 1 —see (5), (6) below o he necessa y defini ions: “Le u∈H2 loc(Ω),n≥3, be a solu ion o (1) |∆u(x)|≤|V(x)u(x)|,x∈Ω, and Ωa connec ed, open subse o Rn. Then he e exis s an >0, depending jus on pand n, such ha i V∈Fp loc =L2,p,VL2,p ≤, p > n−2 2, and u anishes in an open subdomain o Ω, hen umus be ze o e e y- whe e in Ω”. Un o una ely ou p oo happens o be inco ec . The heo em is ne e heless ue, o T. Wolff ob ained a closely ela ed s a emen by using diffe en a gumen s, see [W]. Bo h au ho s suppo ed in pa by Spanish DGICYT g an s. 406 A. Ruiz, L. Vega Ou app oach o unique con inua ion was based upon he ollowing Ca leman es ima e: “The e exis s a cons an C>0such ha o Vin Fp,p>n−2 2 (2) eτxnuL2(V)≤CVL2,p eτxn∆uL2(V−1), holds o e e y uin C∞ 0and τin R”. To ob ain his inequali y we ook a global pa ame ix o he ope a o eτxn∆e−τxn, which la e we ealized can no be uni o mly bounded in τ o V∈L2,p,p≤(n−1)/2 (one has o mul iply he igh hand side a leas by log τ). In ac , he lemma in page 294 o [RV1] gi es he ollowing es ima es o a dyadic decomposi ion Tδo ha pa ame ix : (3) Tδ L2(V)≤Cδ|log δ|VL2,p0 L2(V−1),i p0=n−2 2. (4) Tδ L2(V)≤Cδ1+VL2,p  L2(V−1),i p>n−2 2. Es ima e (3) is ue, bu (4) holds only o p>(n−1)/2. In ac om (3) and i (n−2)/2≤p≤(n−1)/2 a loga i hmic g ow h o he ype Cδ|log δ|is easily ob ained. The in e es ing ema k is ha his g ow h u ns ou o be also necessa y and hence, he e is no con exi y o he bounds o he ope a o Tδin he ange (n−1)/2≤p<(n−2)/2. This ac has some consequences abou he in e pola ion p ope ies in Mo ey spaces ha we shall conside in Sec ion 2. I we subs i u e in (2) he Ca leman weigh τxnby τ(xn+x2 n/2), we can use ou app oach, as we did in [RV2], o imp o e he known esul s on unique con inua ion o solu ions o he inequali y (1) when V∈Lα,p,α<2 —see (5), (6) below o he defini ion. In any case we can no eco e Wolff’s esul (case α= 2) bu only a weake esul o a loga i hmic subs i u e o he space L2,p,p>(n−2)/2. We do no wan o ge in ol ed in hese calcula ions in he p esen no e. On he o he hand we do no know i he inequali y (2) is ue o alse. 2. In e pola ion and Mo ey-Campana o spaces. Mo ey-Campana o classes o m a wo pa ame e amily o spaces Lα,p,α∈(−1, n/p], p∈[1,∞). We say ha ∈L α,p,i is in Lp loc and he e exis s a cons an C>0, which depends on , such ha o e e y x∈Rnand e e y >0, we can find a numbe σ∈R, which depends on ,x, and such ha (3) α −nQ(x, ) | (y)−σ|pdy1/p <C, Co igenda and in e pola ion 407 whe e Q(x, ) is a cube cen e ed a xand olume n. The case α>0 was in oduced by Mo ey in he s udy o he egula i y p oblem o he Calculus o Va ia ions. I was p o ed by Campana o ha σcan be aken ze o wi hou loss o gene ali y —see [C1]. Then o α>0 and ∈L α,p we define (4)  Lα,p = sup x sup α −nQ(x, ) | (y)|pdy1/p . No ice ha i α>0 and p=n/α,p≥1 we ob ain he Lebesgue space Lp. The class was ex ended by Campana o o α≤0 and he [C2] and Mey- e s [M] p o ed o be he space o (−α)-Holde con inuous unc ions; his heo em is known as he in eg al cha ac e iza ion o Holde -con inuous unc ions. When α= 0 we ha e John-Ni embe g space BMO. No ice ha in his case, i.e. α≤0, he space does no change wi h pi αis fixed. Recen ly Mo ey classes, i.e. α>0, ha e been he objec o some wo ks on weigh ed Sobole es ima es, unique con inua ion p ope ies —see [FP], [CS], [ChR], [W], [RV2], and some o he p oblems in PDE —see [RV3], and [T] o example. In his no e we a e conce ned wi h in e pola ion p ope ies o Mo ey spaces o α>0. In pa icula we p o e he lack o he con exi y which cha ac e izes in e pola ion unc o s o exponen θ—see [BL, p. 27]. In pa icula he complex and eal me hods ha e his p ope y. The in e pola ion p ope ies o Mo ey-Campana o spaces ha e been s udied in se e al wo ks du ing he 60’s —see [S], [P] and he e e ences he e in. In pa icula , S ampacchia, [S], and Campana o and Mu hy, [CM] p o ed ha o Ta linea ope a o , and TLqi→Lαi,pi=Ki, wi h 1 ≤pi,qi≤∞,αi>0, i=1,2, hen Tis bounded om Lqθ o Lαθ,pθwi h no m a mos KK1−θ 1Kθ 2, whe e 1 pθ=(1−θ)1 p1+θ1 p2, 1 qθ=(1−θ)1 q1+θ1 q2, and αθ=(1−θ)α1+θα2, and Kdepending jus on θ,αi,pi,qi,i=1,2. In he mo e gene al se ing o Mo ey-Campana o classes, S ein and Zygmund, [S Z], cons uc ed a linea ope a o bounded om Lα,p o Lα,p o some α<0 and om L2 o L2, which is no bounded om BMO o BMO. Le us ema k ha BMO = L0,p and L2=Ln/2,2. Hence in e pola ion h ough he line α= 0 does no hold. We ha e he ollowing esul . 408 A. Ruiz, L. Vega Theo em. Se n>1and 0<α<n. Then, gi en any p1,p2,p3, and C>0, such ha 1≤p2<p 3<n−1 α<p 1<∞, and C>0, he e exis s a con inuous linea ope a o T:Lα,pi→L1,i=1,2,3, such ha TLα,pi→L1≤Ki, i=1,2and (4) TLα,p3→L1>CK 1−θ 1Kθ 2, o 1 p3=(1−θ)1 p1+θ1 p2. P oo o he heo em: Le φbeaC∞ 0non nega i e unc ion such ha φ≤1, φ(x)=1 i |x|<1/4, and φ(x)=0i |x|>1/2. Fo 0 <δ<1 conside he ope a o Tgi en by mul iplica ion by Φδ(x)=φ(|x|)φ(δxn), wi h x=(x1,... ,x n−1): T (x)=Φ δ(x) (x). Then, |T |=Φδ| | ≤δ−(1−1/p)|Φδ |p1/p ≤δ−(1−1/p)δα−n/p Lα,p . And also |T |= νQν Φδ| |, whe e Qνis a collec ion o δ−1cubes o olume one. Hence, |T |≤δ−1 Lα,p. The e o e TLα,p→L1≤δ−1+α−(n−1)/p,i p≥(n−1)/α δ−1,i 1 ≤p≤(n−1)/α. Now ake =Φ δ(x). Then he e exis s a dimensional cons an cn such ha |T |≥cnδ−1, Co igenda and in e pola ion 409 and  Lα,p =c−1 nδ−α+(n−1)/p,i p≥(n−1)/α c−1 n,i 1 ≤p≤(n−1)/α. The e o e TLα,p→L1≥c2 nδ−1,i 1 ≤p≤(n−1)/α. Fix α>0 and ake p1and p2such ha (n−1)/α ≤p1≤n/α, and 1≤p2≤(n−1)/α. Then, on one hand, o θ∈(0,1), we ha e Tθ Lα,p1→L1T1−θ Lα,p2→L1≤δ−1+, whe e =(α−(n−1)/p1)θ>0. On he o he hand o p2<p 3<(n−1)/α,weha e TLα,p3→L1≥c2 nδ−1. Taking δsmall enough we ha e p o ed he heo em. Final ema ks. The abo e heo em can be ex ended o a mo e gene al si ua ion. In pa icula he es ic ion on he dimension, α, and pscan be a oided. In ac we ha e an example o a bounded linea ope a o which is unbounded in a gi en in e media e space. The e o e Mo ey spaces a e no closed by in e pola ion in a s ong o m, and no jus by he lack o con exi y. W i ing his example would ha e made his no e oo la ge and ge ing us oo a om he ini ial pu pose which is he co igenda o ou p e ious pape . On he o he hand he coun e example gi en in he heo em, is a small a ia ion o he one needed o he inequali y (4). We ha e p e e ed o w i e i in his way, o illus a e ha he lack o con exi y is due o he pa icula s uc u e o Mo ey spaces and hei bad beha iou wi h espec o in e pola ion. De ails abou he abo e ques ions will appea elsewhe e. Finally we would like o hank J. Pee e o sha ing wi h us his belie ha Mo ey spaces a e no closed unde in e pola ion. 410 A. Ruiz, L. Vega Re e ences [BL] Be gh, J. and Lo s om, J.,“In e pola ion Spaces,” Sp inge - Ve lag, New Yo k, 1976. [C1] Campana o, S., P op ie a di una amiglia di spazi unzionali, Ann. Scuola N. Sup. Pisa 18 (1964), 137–160. [C2] Campana o, S, P op ie a di h¨olde iani a di alcune classi de unzioni, Ann. Scuola N. Sup. Pisa 17 (1963), 175–188. [CM] Campana o, S. and Mu hy, M. K. V., Una gene alizzazione del eo emi de Riesz–Tho in, Ann. Scuola N. Sup. Pisa 19 (1965), 87–100. [CS] Chanillo, S. and Sawye , E., Unique con inua ion o ∆+ V and he C. Feffe man-Phong class, T ans. AMS 318(1) (1990), 275–300. [ChR] Chia enza, F. and Ruiz, A., Uni o m L2-weigh ed Sobole inequali ies, P oc. AMS 112(1) (1991), 53–64. [FP] Fe e man, C. and Phong, D. H., Lowe bounds o Sch odinge equa ions, J. Eq. aux De i ees Pa ielles, Sain Jean de Mon s, Soc. Ma . de F ance (1982). [M] Meye s, G. N., Mean oscilla ion o e cubes and Holde con inu- i y, P oc. AMS. 15 (1964), 717–721. [P] Pee e, J., On he heo y o Lp,λ spaces, Jou nal o Func ional Analysis 4(1969), 71–87. [RV1] Ruiz, A. and Vega, L., Unique con inua ion o Sch odinge ope a o s in Mo ey spaces, Publicacions Ma ema iques 35 (1991), 291–298. [RV2] Ruiz, A. and Vega, L., Unique con inua ion o he solu ions o he Laplacian plus a d i , Ann. Ins. Fou ie , G enoble 41(3) (1991), 651–663. [RV3] Ruiz, A. and Vega, L., Local egula i y o solu ions o wa e equa ions wi h ime-dependen po en ials, Duke Ma h. J. 76(3) (1994), 913–940. [S] S ampacchia, G.,L(p,λ)- Spaces and in e pola ion, Comm. in Pu e and App. Ma h. 17 (1964), 293–306. [S Z] S ein, E. M. and Zygmund, A., Boundedness o ansla ion in a ian ope a o s on Holde spaces and Lp-spaces, Ann. Ma h. 85 (1967), 337–349. [T] Taylo , M., Analysis on Mo ey spaces and applica ions o Na ie -S okes and o he e olu ion equa ions, Comm. in PDE. 17 (1992), 1407–1456. Co igenda and in e pola ion 411 [W] Wol , T., Unique con inua ion o |∆u|≤V|∇u|and e- la ed p oblems, Re is a Ma em´a ica Ibe oame icana 6(3) (1990), 155–200. 1991 Ma hema ics subjec classifica ions: 42B20, 42B25 Albe o Ruiz: Depa amen o de Ma em´a icas Uni e sidad Au ´onoma de Mad id 28049 Mad id SPAIN e-mail: [email p o ec ed] Luis Vega: Depa amen o de Ma em´a icas Uni e sidad del Pa´ıs Vasco Apa ado 644 48080 Bilbao SPAIN e-mail: m p [email p o ec ed]u.es P ime a e si´o ebuda el 19 de Gene de 1995, da e a e si´o ebuda el 19 de Se emb e de 1995