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