Sobolev meets Poincark

We prove that a very weak form of the PoincarC inequality implies a Sobolev-Poincare ineq~iality in the abstract setting of metric spaces. Sobolev rencontre Poincare Version jkzngaise abrigie L e but de cette Note est la demonstration du theoreme 1 qui affirme que, dans le cadre tres general des espaces metriques, une inegalite fiaible de Poincare entraine une inegalite de Sobole\--Poincare. Ce resultat dont la preuve est elementaire a un certain nombre d'applications; parmi ses corollaires simples on trouve le theoreme de Saloff-Coste [I], theorem 2.1 et une caracterisation utile des puiils p-admissibles [ 2 ] . D e plus, il )a des relations interessantes du theoreme 1 a\-ec les resultats recents de Jerrison [3] et Franchi, Gutierrez et n-heeden [4]. Soit X un espace metrique. Nous disons que 12 c X remplit la condition de chaine C (A. LllI) avec A. 31 > 1 s'il existe une boule fixee Bo C (2 telle que pour chaque .r. E fl on peut trouver une suite de boules Bo B1. L12. d\eL les trois proprietes sunantes 1 AB, C R pour I = 0. 1 2 et B, est centree en I pour tout I suffisamment grand 2 Pour 1 > 0 B, est de ra!on 7 , (dlani 12) 2-' < I , < ilI ((liar11 R) 2 I 3 Pour tout 1 > 0, 11 T a une boule R, c B, n B,+l telle que B, U B,+l c MR, THEOREME 1. S o i t 12 E C (A, 1\1) un sous-eizscmble d 'un espa1.c. mitriqzle X . Adnzettons que 1~1 nzesure / I ilq'finie sur X a la propriiti d~ douhlement, /L ( 2 B ) < Ccl / L (B): Ll = Ll ( .r . r ) , .r: E 12, et r. < 5 tliarrrC2. ~4dmet tons encore que pour des ji~nctiolzs g > 0. g E L". 0 < p < x, u E L:<,,. (12, / I ) / ( / rersion ilhstrliite ile l 'ine~yaliti . f i ~ b l e de Polncare, soit ~>t;l.$ie pozu clzilqlte boule B telle qzie XB C R. A l u u il esiste k > 1 p i ne dipend qzte de p et C,,, et une i,uizstnnte C2 = C2 (C1. Cd. 11. k r A. -II) telle q1Oun i ~ i t l ' inegalitt globnle i/e Soholez,-Poinral.c;c suirante Sz k p < 1. on renzpla~e ~ L Q par uo , . MAIN RESULT. The purpose of this Note is the proof of theorem 1 which, roughly speaking. states that, in the very general setting of metric spaces. a weak Poincar6 inequality Note presentee par Hai'm BREZIS. 0764-4442/95/0320 121 1 $ 2.00 O Academe des Sc~ences 1212 P. Hajtasz and P. Koskela implies a Sobolev-Poincark inequality. This result has a number of applications and its proof is surprisingly elementary. In particular. the theorem of Saloff-Coste ([I]. theorem 2.1) is a very special case of theorem 1: also it provides an elementary characterization of 11-admissible weights [2] and it simplifies and extends some of the results by Jerison 131. and Franchi. Gutierrez and Wheeden [4]. Let X be a metric space. We say that 12 c X satisfies the chain condition C'(X. *\I). where A. 111 2 1. if there exists a distinguished ball B,, C 12 such that for every .I: E 12 there exists an infinite sequence of balls Bo. B1. B2. ... (called "chain") with the following properties. 1. XB, c (2 for i = 0. 1 , 2. ... and R, is centered at .I: for all sufficiently large i . 2. For i 2 0 the radius r. , of B, satisties 121-I (tliarn ( 2 ) 2-I < r., < (dial11 (2) 2 ' . 3. For every .I > (1 there is a ball R, c B, n B,+l such that B, U B,+l c AIR, . Here and in what follows. by B we always denote a ball and by tB. where t > 0. a ball concentric with B and with radius t times that of B. By C we denote a general constant which can change its value even in a single line. The above chain condition is different from the cornmonly used Boman's chain condition (cf: 14)). If Q c R" is a bounded domain with smooth boundary then it satisfies the C (A. AI) condition for all X 2 I . The following lemma and its corollary provide us with more sophisticated examples. LEMMA 1. Let (X, (1) he a rnetric space s~rch tlqczt bo~rtzded and closed set.s elre conzpact. A s s ~ l ~ i e tlzrrt the ~ i e t r i c (1 1zcr.s tlze propert! that for ever-! t ~ . o poi lit.^ O . h € ,y the di.sturzc.e ( / ( ( I . b ) is eyilul to tlze injirnirrn of the lengths qf ccontinuou.s ci1n.e~ that joirz c1 and h (in pnrticular L1.e assunie that sirch u cirrlle crI~va?..s exists). Tlzen tliere exists u slzortest path y ,from (7 to b. This cur\*e he1.s the ,follo~~~iri,q segnlent propertj. For el.erT z E y, (1 (a. h ) = d ( n . a ) + d ( a . b ) . This lemma is due to Busemann [5]. p. 25 (cf: [4]. p. 592). COROLLARY 1. Fix X > 1. Let the metric space (S. (1) firlfill the h p o t h e s i . ~ of the lenzrnn above. Then rlJer:\. blill B C S .satisjie.s the C (A. 211) conclitiorz rt.it11 ci certcrirl M ~vhicli depe1id.s on the choice qf' A. The main result of this Note reads as follows. THEOREM 1 . Let 12 C X. (1 E C (A. 31). A.ssirine that / I is c1 cio~lbling tnerisi~re: / I ( 2 B) 5 C:, / r ( B ) 1.17henever B = B ( : r . I . ) . .r E a, I . < 5 diain Q,,. Ass~rnir ~ J I C L ~ > 0, g E L1' ( i t , / L ) . (1 < p < x, u E L:,,,. ( ( 2 % 1 1 ) (Ire .such that the,follo~~,irzg abstnrct ~,er.sion o f the local wectk Poirlcare ineq~lality holds: \thenever XB c 1 2 arzcl r 1s tlze radius of the bull B Then there exist5 A. > 1 I . I ~ K / I depends on 11 ar~d the c/oublirzg constant C,, on/\, alld C'] = C2 (C1. C,,. 1). X . A. 111) ~ L K I Z thcit the follou zng plobc~l so hole^^-Pornt crrP irzequallt~ holcls If Xp < 1. we replace u s ) by u ~ , , . Here and in what follows u~ = Z L d p . Sobolev meets PoincarC 1213 R e m ~ ~ r k . If we know in addition that 6,, = log, C,! < p. then we can prove more. Namely. for ho = J). we get exponential integrability. and. for < 1). Holder continuity of u . Moreover. there is a two-weighted version of the above theorem where integration in the different sides in ( 1 ) and (2) is with respect to different measures. see [6]. Prooj: It suffices to assume u*,, = 0 and estimate ( f ll l k j l ) ' /"" . Let :r E At = < 0 (1111 > f ) be a Lebesgue point of 11. Let Bo. B1. B2. ... be a chain assigned to .I:. We have j ~ l ~ , A 111 (.r) j > t . ~ L B ~ , = 0 . Using the doubling property and PoincarC inequality ( 1 ) we compute