Erratum to Multipliers and Morrey spaces.

Transcription

1 Erratum to Multipliers Morrey spaces. Pierre Gilles Lemarié Rieusset Abstract We correct the complex interpolation results for Morrey spaces which is false for the first interpolation functor of Calderón, but is exact for the Calderón s second interpolation functor. Keywords : Morrey spaces; interpolation. 200 Mathematics Subject Classification : 42B35 In my paper Multipliers Morrey spaces [3], there is a slight mistake in Theorem 3, concerning the interpolation of Morrey spaces. Let B be the collection of all Euclidean balls B on R d : B = B(x B, r B = {x R d / x x B < R B. For B B, we write B = dx = B(0, B rd B. Define, for < p q < +, the space Ṁ p,q (R n as the space of locally p-integrable functions f such that f Ṁ p,q := sup B /q /p ( B B f(x p dx /p < +. It is easy to check that, when < p 0 q 0 < + < p q < +, for some θ (0,, then Ṁ p 0,q 0 Ṁ p,q f Ṁ p,q B f θ Ṁ p 0,q 0 f θ Ṁ p,q The question I studied was then whether one may find an interpolation functor F of exponent θ such that F (Ṁ p 0,q 0 = Ṁ p,q. If so, one should have the continuous embeddings ] θ, Ṁ p,q ] θ,. Laboratoire de Mathématiques et Modélisation d Évry, UMR CNRS 807, Université d Évry; plemarie@univ-evry.fr

2 The inclusion [A 0, A ] θ, F (A 0, A is a direct conclusion from the inequality f F (A0,A C f θ A 0 f θ A (obtained by interpolation inequalities for the operator norms of λ λf from R to A 0 from R to A, hence from R to F (A 0, A. The inclusion F (A 0, A [A 0, A ] θ, is proven in [] under the assumption that A 0 A is dense in A 0 in A. This is not the case for Morrey spaces. However, one may easily adapt the proof, as Morrey spaces are dual spaces (see for instance [6]. If we assume that A 0 = B 0 A = B that B 0 B is dense in B 0 B ; then, for b B 0 B, the linear form T b : f f b has a norm less than b B0 as an operator from A 0 to R less than b B as an operator from A to R, hence as a norm less than C b θ B 0 b θ B ; thus, T f : b f b is a continuous linear form on [B 0, B ] θ,. This gives that f ([B 0, B ] θ, = [B 0, B ] θ, (since B 0 B is dense in B 0 B. The theorem I proved in [3] is the following one: Theorem Let < p 0 q 0 < + < p q < +, for some θ (0,. Then there exists an interpolation functor F of exponent θ such that F (Ṁ p 0,q 0 = Ṁ p,q if only if p 0 /q 0 = p /q. The negative result for the case p 0 /q 0 p /q was proven by a generalization of a counterexample by Ruiz Vega [4] which proves that, in that case, we don t have the embedding of Ṁ p,q into ] θ,. The proof for the positive result (on the case p 0 /q 0 = p /q was inexact. I claimed that in that case we have the complex interpolation Ṁ p,q = ] θ. But this is false as pointed to me by Sickel (who has recently characterized the intermediate space Ṁ p,q ] θ in a joint work with Yang Yuan [5]. Indeed, it is easy to see that, when p 0 /q 0 = p /q = p 0 /q 0 < p 0 p, Ṁ p 0,q 0 is not dense in Ṁ p,q, while it is always true that A 0 A is dense in [A 0, A ] θ (see []. Sickel s counterexample is very clear : if r = min(p 0, p s = max(q 0, q, we have Ṁ p,q Ṁ r,q Ṁ p 0,q 0 Ṁ r,s ; thus the applications f ρ d(/q /r B(0,ρ f are equicontinuous from Ṁ p,q to L r ; for f Ṁ p 0,q 0, we have lim ρ 0 ρ d(/q /r B(0,ρ f r = 0, while for f 0 = x d/q Ṁ p,q, we have lim ρ 0 ρ d(/q /r B(0,ρ f 0 r > 0; thus f 0 does not belong to the closure of Ṁ p 0,q 0 2

3 However, a slight modification of the proof of [3] gives the following theorem : Theorem 2 Let < p 0 q 0 < + < p q < +, If p 0 /q 0 = p /q, then Ṁ p,q = ] θ. Let us recall that Calderón [2] defined two complex interpolation functors : [A 0, A ] θ [A 0, A ] θ. We have [A 0, A ] θ = [A 0, A ] θ (with equality when at least one of the two spaces A 0 A is reflexive. Proof : Let us recall the definition of [A 0, A ] θ [A 0, A ] θ. Let Ω be the open complex strip Ω = {z C / 0 < Rz < }. F is the space of functions F defined on the closed complex strip Ω such that : Then. F is continuous bounded from Ω to A 0 + A 2. F is analytic from Ω to A 0 + A 3. t F (it is continuous from R to A 0, lim t + F (it A0 = 0 4. t F (+it is continuous from R to A, lim t + F (+it A0 = 0 f [A 0, A ] θ F F, f = F (θ f [A0,A ] θ = inf max(sup F (it A0, sup F ( + it A. f=f (θ t R t R On the other h, G is the space of functions G defined on the closed complex strip Ω such that :. + z G is continuous bounded from Ω to A 0 + A 2. G is analytic from Ω to A 0 + A 3. t G(it G(0 is Lipschitz from R to A 0 3

4 4. t G( + it G( is Lipschitz from R to A Then f [A 0, A ] θ G G, f = G (θ f [A0,A ] = inf max( sup G(it 2 G(it θ A0, sup G( + it 2 G( + it A. f=g (θ t,t 2 R t 2 t t,t 2 R t 2 t Let us remark that, for continuous functions, (strong analyticity is equivalent to weak analyticiy or even *-weak analyticity when A 0 A are dual spaces of B 0 B with B 0 B dense in B 0 B. Indeed, analyticity is equivalent to the fact that, whenever the closed ball B(z0, r is contained in Ω w z 0 < r, then F (w = we have, for b B 0 B, b F (z z w B 0 B,A 0 +A = F (z z w. As F is continuous, b F (z B0 B,A 0 +A z w The equvalence remains true for *-weakly continuous functions. However, of course, there is no equivalence between (strong continuity *-weak continuity. In the original proof of [3], one made two remaks :. Let < p 0 q 0 < + < p q < +,. If F is an interpolation functor of exponent θ that satisfies F (L p 0, L p = L p, then F (Ṁ p 0,q 0 Ṁ p,q. Thus, we have the embeddings of ] θ,p, ] θ ] θ into Ṁ p,q. 2. When moreover p 0 /q 0 = p /q = p/q we may define for f Ṁ p,q the function F (z = f p f f ( z +z p p 0 p. This is a bounded *-weakly continuous function of z = x+iy (for 0 x with values in Ṁ p 0,q 0 +Ṁ p,q, holomorphic on the strip 0 < x <, with sup R F (iy Ṁ p 0,q 0 < +, sup R F ( + iy Ṁ p,q < +, F (θ = f. If F was strongly continuous, we would find that f = F (θ would belong to ] θ. But F is only *-weakly continuous. We may define G(z = z F (w dw. Then we have G G, 0 G (θ = f; thus f belongs to ] θ 4

5 References [] J. Bergh J. Löfström. Interpolation spaces. Springer-Verlag, 976. [2] A.P. Calderón. Intermediate spaces interpolation: the complex method. Studia Math., 24:3 90, 964. [3] P.G. Lemarié-Rieusset. Multipliers Morrey spaces. Potential Analysis, 38:74 752, 203. [4] A. Ruiz L. Vega. Corrigenda to unique continuation for Schrödinger operators a remark on interpolation of Morrey spaces. Publ. Mat., 39:404 4, 995. [5] W. Sickel. Personnal communication, march 204. [6] C.T. Zorko. Morrey spaces. Proc. Amer. Math. Soc., 98: ,