On the Calderón-Zygmund lemma for Sobolev functions

Transcription

1 arxv: v1 [ath.ca] 28 Oct 2008 On the Calderón-Zygund lea for Sobolev functons Pascal Auscher october 16, 2008 Abstract We correct an naccuracy n the proof of a result n [Aus1] MSC: 42B20, 46E35 Key words: Calderón-Zygund decoposton; Sobolev spaces. We recall the lea. Lea 0.1. Let n 1, 1 p and f D (R n ) be such that f p <. Let α > 0. Then, one can fnd a collecton of cubes (Q ), functons g and b such that (0.1) f = g + and the followng propertes hold: (0.2) g Cα, (0.3) b W 1,p 0 (Q ) and b Q b p Cα p Q, (0.4) Q Cα R p f p, n Unversté Pars-Sud and CNRS UMR 8628, Departent de Mathéatques, Orsay Cedex (France) Eal: pascal.auscher@ath.u-psud.fr 1

2 (0.5) 1 Q N, where C and N depend only on denson and p. Ths lea was frst stated n [Aus1] n R n. Then t appears n varous fors and extensons n [Aus2] (sae proof), [AC] (sae proof on anfolds), [AM] (on R n but wth a doublng weght), B. Ben Al s PhD thess [Be] and [AB] (The Sobolev space s odfed to adapt to Schrödnger operators), N. Badr s PhD thess [Ba] and [Ba1, Ba2] (used toward nterpolaton of Sobolev spaces on anfolds and easured etrc spaces) and n [BR] (Sobolev spaces on graphs). The sae naccuracy can be corrected everywhere as below. The proof of the generalsaton to hgher order Sobolev spaces n [Aus1] can also be corrected wth slar deas. The second equaton tells that g s n fact Lpschtz contnuous. There s a drect proof of ths fact n N. Badr s thess [Ba]. The proof proposed n [Aus1] s as follows: Defne b = (f c )X where c are approprate nubers and (X ) fors a sooth partton of unty of Ω = Q subordnated to the cubes ( 1 Q 2 ) wth support of X contaned n Q. For exaple, the choce c = f(x ) for soe well chosen x or c = Q f, the ean of f over the cube Q, ensures that b l 1 s locally ntegrable (l s the length of Q ) and that b s a dstrbuton on R n. Then g defned as g = f b s a dstrbuton on R n. Its gradent g can be calculated as g = ( f)1 F + h n the sense of dstrbutons (on R n ) wth h = c X. It s then a consequence of the constructon of the set F = Ω c that f s bounded on F by α and then t s shown that h s bounded by Cα, whch ples the boundedness of g. Everythng s correct n the arguent above BUT the representaton of h. The seres c X, vewed as the dstrbutonal dervatve on R n of c X, ay not be a easurable functon (secton) on R n. For exaple, f c = 1 for all, then X = 1 Ω s a non-zero dstrbuton supported on the boundary of Ω (a easure f Ω has locally fnte pereter). One needs to renoralze the seres to ake t converge n the dstrbuton sense. Here we gve correct renoralzatons of h. A frst one s obtaned rght away fro dfferentaton of g: h = (f c ) X. The convergence n the dstrbutonal sense n R n s n fact hdden of [Aus1]. 2

3 A second one s h = ( ) (c c ) X X. Ths representaton converges n the dstrbutonal sense n R n and can be shown to be a bounded functon. Let us show how to obtan the second representaton n the sense of dstrbutons. Then the proof of boundedness s as n [Aus1]. Take a test functon φ n R n. Then by defnton the dstrbuton b tested aganst φ s gven by b φ. To copute ths, we take a fnte subset J of the set I of ndces and we have to pass to the lt n the su restrcted to J as J I. Because now the su s fnte, and all functons have support n the set Ω, we can ntroduce X = 1 Ω. We have b φ = b X φ. J Now recall that b = (f c )X. Call I the set of ndces such that the support of X eets the support of X. By property of the Whtney cubes, I s a fnte set wth bounded cardnal. Hence we can wrte b X φ = f X X φ+ (f c ) X X φ. J J I J J I It s clear that the frst su n the RHS converges to f φ as J I. As Ω for the second t s equal to (c c ) X X φ + (f c ) X X φ J I J I As one can show (wth the arguent n [Aus1]) that c c X X Cα 3

4 one has l J I J I (c c ) X X φ = where h s defned above. Fnally, wrte (f c ) X X φ = wth J I R,J = J I X. h φ b R,J φ By constructon of the X and propertes of Whtney cubes, X Cl 1 I where l s the length of Q, and on the support of X X = X = 0. I I As b l 1 has been shown to be locally ntegrable, one can conclude by the Lebesgue donated convergence theore that References l J I b R,J φ = 0. [Aus1] P. Auscher. On L p -estates for square roots of second order ellptc operators on R n, Publ. Mat. 48 (2004), [Aus2] P. Auscher, On necessary and suffcent condtons for L p estates of Resz transfor assocated ellptc operators on R n and related estates, Meors Aer. Math. Soc 186 (871) (2007). [AB] P. Auscher & B. Ben Al, Maxal nequaltes and Resz transfor estates on L p spaces for Schrödnger operators wth nonnegatve potentals, Ann. Inst. Fourer 57 no. 6 (2007),

5 [AC] P. Auscher & T. Coulhon, Resz transfors on anfolds and Poncaré nequaltes, Ann. Sc. Nor. Super. Psa Cl. Sc. (5) 4 (2005), [AM] P. Auscher & J.M. Martell, Weghted nor nequaltes, off-dagonal estates and ellptc operators. Part I: General operator theory and weghts, Adv. Math. 212 (2007), no. 1, [Ba] Nadne Badr, PhD thess, Unversty of Orsay, [Ba1] Nadne Badr, Real nterpolaton of Sobolev spaces, arxv: , to appear n Math. Scand. [Ba2] Nadne Badr, Real nterpolaton of Sobolev spaces assocated to a weght, arxv: [BR] N. Badr, E. Russ, Interpolaton of Sobolev spaces, Lttlewood-Paley nequaltes and Resz transfors on graphs, arxv: v1. To appear n Pub. Mat. [Be] Besa Ben Al, PhD thess, Unversty of Orsay,