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