CORRIGENDA TO “ UNIQUE CONTINUATION FOR SCHRÖDINGER OPERATORS ” AND A REMARK ON INTERPOLATION OF MORREY SPACES

The purpose of this note is twofold. First it is a corrigenda of our paper [RV1]. And secondly we make some remarks concerning the interpolation properties of Morrey spaces. 1. Corrigenda. In our paper “Unique continuation for Schrödinger 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 ∈ H loc(Ω), n ≥ 3, be a solution of (1) |∆u(x)| ≤ |V (x)u(x)|, x ∈ Ω, and Ω a connected, open subset of R. Then there exists an > 0, depending just on p and n, such that if V ∈ F p loc = L, ‖V ‖L2,p ≤ , p > n− 2 2 , and u vanishes in an open subdomain of Ω, then u must 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 > 0 such that for V in F , p > n−2 2 (2) ‖eu‖L2(V ) ≤ C‖V ‖L2,p‖e∆u‖L2(V −1), holds for every u in C∞ 0 and τ in 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 τ for V ∈ L, 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 Tδ of 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 ) ≤ Cδ ‖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 Tδ in 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 τxn by τ(xn + xn/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 ∈ L, 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 L, 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 ∈ Lα,p, if f is in Lploc and there exists a constant C > 0, which depends on f , such that for every x ∈ R and every r > 0, we can find a number σ ∈ R, which depends on f , x, and r such that


Corrigenda.
In our paper "Unique continuation for Schrödinger 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 ∈ H 2 loc (Ω), n ≥ 3, be a solution of and Ω a connected, open subset of R n .Then there exists an > 0, depending just on p and n, such that if and u vanishes in an open subdomain of Ω, then u must 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.

Our approach to unique continuation was based upon the following Carleman estimate:
"There exists a constant C > 0 such that for V in F p , p > n−2

2
(2) holds for every u in C ∞ 0 and τ in R".To obtain this inequality we took a global parametrix of the operator e τxn ∆e −τxn , which later we realized can not be uniformly bounded in τ for V ∈ L 2,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 T δ of that parametrix : (3) (4) 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 T δ in 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 τ x n by τ (x n + x 2 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 ∈ L α,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 L 2,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.

Interpolation and Morrey-Campanato spaces.
Morrey-Campanato classes form a two parameter family of spaces loc and there exists a constant C > 0, which depends on f , such that for every x ∈ R n and every r > 0, we can find a number σ ∈ R, which depends on f , x, and r such that where Q(x, r) is a cube centered at x and volume r n .
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 σ can be taken zero without loss of generality -see [C1].Then for α > 0 and f ∈ L α,p we define (4) Notice that if α > 0 and p = n/α, p ≥ 1 we obtain the Lebesgue space L p .
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 p if α is fixed.
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 T a linear operator, and , and α θ = (1 − θ)α 1 + θα 2 , and K depending just on θ, α i , p i , q i , i = 1, 2.
In the more general setting of Morrey-Campanato classes, Stein and Zygmund, [StZ], constructed a linear operator bounded from L α,p to L α,p for some α < 0 and from L 2 to L 2 , which is not bounded from BMO to BMO.Let us remark that BMO = L 0,p and L 2 = L n/2,2 .Hence interpolation through the line α = 0 does not hold.
We have the following result.
On the other hand for p 2 < p 3 < (n − 1)/α, we have Taking δ small 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 s can 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.