3 New Calderón-Zygmund decompositions for Sobolev functions. 7

Transcription

1 New Calderón-Zygmund decomposton for Sobolev functons N. Badr Insttut Camlle Jordan Unversté Claude Bernard Lyon 43 boulevard du Novembre 98 F Vlleurbanne Cedex F. Berncot Unversté de Pars-Sud F-9405 Orsay Cedex Aprl 2, 200 Abstract We state a new Calderón-Zygmund decomposton for Sobolev spaces on a doublng Remannan manfold. Our hypotheses are weaker than those of the already known decomposton whch used classcal Poncaré nequaltes. Key-words : Calderón-Zygmund decomposton, Sobolev spaces, Poncaré nequaltes. MSC : 42B20, 46E35. Contents Introducton 2 2 Prelmnares 3 2. The doublng property Classcal Poncaré nequalty Estmates for the heat kernel The K-method of real nterpolaton New Calderón-Zygmund decompostons for Sobolev functons Decomposton usng abstract oscllaton operators Applcaton to real Interpolaton of Sobolev spaces Homogeneous verson

2 4 Pseudo-Poncaré nequaltes and Applcatons 5 4. The partcular case of Pseudo-Poncaré Inequaltes Applcaton to Reverse Resz transform nequaltes Applcaton to Gaglardo-Nrenberg nequaltes Introducton The purpose of ths artcle s to weaken assumptons of the already known Calderón- Zygmund decomposton for Sobolev functons. Ths well-known tool was frst stated by P. Auscher n [2]. It exactly corresponds to the Calderón-Zygmund decomposton n a context of Sobolev spaces. Let us brefly recall the deas of such decomposton. In [34], E. Sten stated ths decomposton for Lebesgue spaces as followng. Let (X, d, µ) be a space of homogeneous type and p. Gven a functon f L p (X), the decomposton gves a precse way of parttonng X nto two subsets: one where f s essentally small (bounded n L norm); the other a countable collecton of cubes where f s essentally large, but where some control of the functon s obtaned n L norm. Ths leads to the assocated Calderón-Zygmund decomposton of f, where f s wrtten as the sum of good and bad functons, usng the above subsets. Ths decomposton s a basc tool n Harmonc analyss and the study of sngular ntegrals. One of the applcatons s the followng : an L 2 -bounded Calderón-Zygmund operator s of weak type (, ) and so L p bounded for every p (, ). In [2], P. Auscher extended these deas for Sobolev spaces. followng: Hs decomposton s the Theorem. Let n, p [, ) and f D (R n ) be such that f L p <. Let α > 0. Then, one can fnd a collecton of cubes (Q ), functons g and b such that f = g + b and the followng propertes hold: g L Cα, b W,p 0 (Q ) and b p Cα p Q, Q Q Cα R p f p, n Q N, where C and N depend only on the dmenson n and on p. 2

3 The mportant pont n ths decomposton s the fact that the functons b are supported n the correspondng balls, whle the orgnal Calderón-Zygmund decomposton appled to f would not gve ths. The proof reles on an approprate use of Poncaré nequalty and was then extended to a doublng manfold wth Poncaré nequalty by P. Auscher and T. Coulhon n [6]. Ths decomposton s used n many works and t appears n varous forms and extensons. For example n [6] (same proof on manfolds), [8] (on R n but wth a doublng weght), B. Ben Al s PhD thess [6] and [5], [4] (the Sobolev space s modfed to adapt to Schrödnger operators), N. Badr s PhD thess [9] and [0, ] (used toward nterpolaton of Sobolev spaces on manfolds and measured metrc spaces) and n [3] (Sobolev spaces on graphs). The am of ths artcle s to extend the proof usng other knd of Poncaré nequaltes. Ths work can be ntegrated n several recent works, where the authors look for replacng the mean-value operators by other ones n the defnton of Hardy spaces for example or n the defnton of maxmal operators (see [9, 20, 26,?, 32]... ). Manly, Secton 3 s devoted to the proof of Calderón-Zygmund decompostons for Sobolev functons (as n Theorem.) n an abstract framework of a doublng Remannan manfold under assumptons nvolvng new knd of Poncaré nequaltes. Then we gve an applcaton to the real nterpolaton of Sobolev spaces W,p. In Secton 4, we focus on a partcular case (usng the heat semgroup) correspondng to the so-called pseudo-poncaré nequaltes. We specfy that these new Poncaré nequaltes are weaker than the classcal ones and permt to nsure the Calderón-Zygmund decomposton for Sobolev functons. We gve some applcatons usng ths mprovement. 2 Prelmnares Throughout ths paper we wll denote by E the characterstc functon of a set E and E c the complement of E. If X s a metrc space, Lp wll be the set of real Lpschtz functons on X and Lp 0 the set of real, compactly supported Lpschtz functons on X. For a ball Q n a metrc space, λq denotes the ball co-centered wth Q and wth radus λ tmes that of Q. Fnally, C wll be a constant that may change from an nequalty to another and we wll use u v to say that there exsts a constant C such that u Cv and u v to say that u v and v u. In all ths paper, M denotes a complete Remannan manfold. We wrte µ for the Remannan measure on M, for the Remannan gradent, for the length on the tangent space (forgettng the subscrpt x for smplcty) and L p for the norm on L p := L p (M, µ), p +. We denote by Q(x, r) the open ball of center x M and radus r > 0. We wll use the postve Laplace-Beltram operator defned by f, g C 0 (M), f, g = f, g. We deal wth the Sobolev spaces of order W,p := W,p (M), where the norm s defned by: f W,p (M) := f p + f L p. 3

4 2. The doublng property Defnton 2. (Doublng property) Let M be a Remannan manfold. One says that M satsfes the doublng property (D) f there exsts a constant C > 0, such that for all x M, r > 0 we have µ(q(x, 2r)) Cµ(Q(x, r)). (D) Lemma 2.2 Let M be a Remannan manfold satsfyng (D) and let d = log 2 C. Then for all x, y M and θ Observe that f M satsfes (D) then µ(q(x, θr)) Cθ d µ(q(x, R)) () dam(m) < µ(m) < (see []). Therefore f M s a complete Remannan manfold satsfyng (D) then µ(m) =. Theorem 2.3 (Maxmal theorem) ([22]) Let M be a Remannan manfold satsfyng (D). Denote by M the uncentered Hardy-Lttlewood maxmal functon over open balls of M defned by where f E := fdµ := E Mf(x) := sup Q ball x Q f Q µ(e) E moreover of weak type (, ). Consequently for s (0, ), the operator M s defned by fdµ. Then for every p (, ], M s L p bounded and M s f(x) := [M( f s )(x)] /s s of weak type (s, s) and L p bounded for all p (s, ]. 2.2 Classcal Poncaré nequalty Defnton 2.4 ( Classcal Poncaré nequalty on M) We say that a complete Remannan manfold M admts a Poncaré nequalty (P q ) for some q [, ) f there exsts a constant C > 0 such that, for every functon f Lp 0 (M) 2 and every ball Q of M of radus r > 0, we have ( /q ( /q f f Q dµ) q Cr f dµ) q. (P q ) Q Remark 2.5 By densty of C 0 (M) n Lp 0 (M), we can replace Lp 0 (M) by C 0 (M). Let us recall some known facts about Poncaré nequaltes wth varyng q. It s known that (P q ) mples (P p ) when p q (see [29]). Thus, f the set of q such that (P q ) holds s not empty, then t s an nterval unbounded on the rght. A recent result of S. Keth and X. Zhong (see [30]) asserts that ths nterval s open n [, + [ : An operator T s of weak type (p, p) f there s C > 0 such that for any α > 0, µ({x; T f(x) > α}) C α f p p. p 2 compaclty supported Lpshtz functon defned on M. Q 4

5 Theorem 2.6 Let (X, d, µ) be a complete metrc-measure space wth µ doublng and admttng a Poncaré nequalty (P q ), for some < q <. Then there exsts ɛ > 0 such that (X, d, µ) admts (P p ) for every p > q ɛ. 2.3 Estmates for the heat kernel We recall the followng off-dagonal decays of the heat semgroup and the lnk between these decays and the boundedness of the Resz transform, the doublng property and Poncaré nequalty. We refer the reader to the work of P. Auscher, T. Coulhon, X. T. Duong and S. Hofmann [7] and [6] for more detals about all these notons and how they are related. Let us consder the followng two nequaltes: f p C( 2 f p + f p ) (nhr p ) and ( 2 f p + f p ) C f p. (nhrr p ) Theorem 2.7 Let M be a complete doublng Remannan manfold. The nequaltes (nhr 2 ) and (nhrr 2 ) are always satsfed. ([23]) Assume that the heat kernel p t of the semgroup e t satsfes the followng pontwse estmate: p t (x, x) µ(b(x, t /2 )). (DUE) Then for all p (, 2], (nhr p ) and (nhrr p ) hold 3. ([28], Theorem.) Under (D), (DU E) self-mproves nto the followng Gaussan upper-bound estmate of p t p t (x, y) d 2 (x,y) µ(b(y, t /2 )) e c t. (UE) Note that (UE) mples (L L ) off-dagonal decays for (e t ) t>0. Under (UE), the collecton ( t e t ) t>0 satsfes L 2 L 2 off-dagonal decays. Under (DUE) and by the analtcty of the heat semgroup, the followng pontwse upper bound for the kernel of e t : t p t t holds (see [25], Theorem 4 and [28], Corollary 3.3): t t p t(x, y) d 2 (x,y) µ(b(y, t /2 )) e c t. (2) Theorem 2.8 ([3, 33]) The conjuncton of (D) and Poncaré nequalty (P 2 ) on M s equvalent to the followng L-Yau nequalty d 2 (x,y) µ(b(y, t /2 )) e c t wth some constants c, c 2 > 0. 3 The assumptons n [23] are even weaker. p t (x, y) d 2 (x,y) µ(b(y, t /2 )) e c 2 t, (LY ) 5

6 Theorem 2.9 ([7]) The L p -boundedness of the Resz transform ( ) /2 mples e t L p L p t. (G p ) Moreover, under (P 2 ) and (G p0 ) wth p 0 > 2, the collecton ( t e t ) t>0 satsfes some (L p L p ) off-dagonal decays for every p [2, p 0 ). Remark 2.0 All these results are proved n ther homogeneous verson, wth homogeneous propertes (R p ) and (RR p ). It s essentally based on the well-known Calderón- Zygmund decomposton for Sobolev functons. Ths tool was extended for non-homogeneous Sobolev spaces (see [0]). Thus by exactly the same proof, we can obtan an analogous non-homogeneous verson and then prove all these results. 2.4 The K-method of real nterpolaton We refer the reader to [7], [8] for detals on the development of ths theory. Here we only recall the essentals to be used n the sequel. Let A 0, A be two normed vector spaces embedded n a topologcal Hausdorff vector space V. For each a A 0 + A and t > 0, we defne the K-functonal of nterpolaton by K(a, t, A 0, A ) = nf a=a 0 +a ( a 0 A0 + t a A ). For 0 < θ <, q, we denote by (A 0, A ) θ,q the nterpolaton space between A 0 and A : { ( (A 0, A ) θ,q = a A 0 + A : a θ,q = (t θ K(a, t, A 0, A )) q dt ) } q <. t It s an exact nterpolaton space of exponent θ between A 0 and A (see [8], Chapter II). Defnton 2. Let f be a measurable functon on a measure space (X, µ). The decreasng rearrangement of f s the functon f defned for every t 0 by f (t) = nf {λ : µ({x : f(x) > λ}) t}. The maxmal decreasng rearrangement of f s the functon f defned for every t > 0 by f (t) = t 0 t 0 f (s)ds. Proposton 2.2 From the propertes of f, we menton:. (f + g) f + g. 2. (Mf) f. 3. µ({x; f(x) > f (t)}) t. 4. p (, ], f p f p. 6

7 We exactly know the functonal K for Lebesgue spaces: Proposton 2.3 Take 0 < p 0 < p. We have: where α = p 0 p. ( t α K(f, t, L p 0, L p ) 0 /p0 ( ) /p [f (s)] p 0 ds) + t [f (s)] p ds, t α 3 New Calderón-Zygmund decompostons for Sobolev functons. In the ntroducton, we recalled the man use of Calderón-Zygmund decompostons for Sobolev functons. In the prevously cted works, ths decomposton reles on Poncaré nequaltes and some trcks wth the mean-value operators. We present here smlar arguments wth abstract operators, requrng new Poncaré nequaltes. Then, we gve some applcatons to real nterpolaton of Sobolev spaces. 3. Decomposton usng abstract oscllaton operators Let A := (A Q ) Q be a collecton of operators (actng from W,p to W,p loc ) ndexed by the balls of the manfold (A Q can be thought to be smlar to the mean operator over the ball Q). Defnton 3. We defne a new maxmal operator assocated to ths collecton: for s p and all functons f W,p M A,s (f)(x) := sup Q; Q x µ(q) /s A Q(f) W,s (Q). Let us now defne the assumptons that we need on the collecton A. Defnton 3.2 ) We say that for q [, ] 4, the manfold M satsfes a Poncaré nequalty (P q ) relatvely to the collecton A f there s a constant C such that for every ball Q (of radus r Q ) and for all functons f W,p ; p q: ( Q f A Q (f) q dµ) /q Cr Q sup s ( sq ( f + f ) q dµ) /q. 2) For q r, we say that the collecton A satsfes L q L r off-dagonal estmates f a. there are constants C > 0 and N N such that for all equvalent balls Q, Q (.e. Q Q NQ) and all functons f W,p ; p q, we have µ(q) /r A Q(f) A Q (f) L r (NQ) C r Q nf NQ M q ( f + f ) (3) 4 we take the supremun nstead of the L q average when q =. 7

8 b. and for every ball Q Here s our man result : µ(q) /r A Q(f) W,r (Q) C nf Q M q ( f + f ). (4) Theorem 3.3 Let M be a complete Remannan manfold satsfyng (D) and of nfnte measure. Consder a collecton A = (A Q ) Q of operators defned on M. Assume that M satsfes the Poncaré nequalty (P q ) relatvely to the collecton A for some q [, ), and that A satsfes L q L r off-dagonal estmates for some r (q, ]. Let q p < r, f W,p and α > 0. Then one can fnd a collecton of balls (Q ), functons g W,r and b W,q wth the followng propertes f = g + b (5) g W,r f p/r W,p α p/r, Q ( g r + g r )dµ α r µ( Q ) (6) supp (b ) Q, b W,q αµ(q ) /q (7) µ(q ) Cα p ( f + f ) p dµ (8) Q N. (9) Remark 3.4 From the assumed L q L r off-dagonal estmates for A and Theorem 2.3, we deduce that the maxmal operator M A,q s contnuous from W,q to L q, and from W,p to L p for p (q, r]. Proof : We follow the deas of [0] where the result s proved for the partcular case A Q (f) := fdµ. Let f W,p and α > 0. Consder the set Ω := {x M; M q ( f + f )(x) + M A,q (f)(x) > α}. We can assume that ths set s non empty (otherwse the result s obvous takng g = f). Wth ths assumpton, the dfferent maxmal operators are of weak type (p, p) so ( ) µ(ω) Cα p f p dµ + f p dµ (0) < +. In partcular Ω M as µ(m) =. Let F be the complement of Ω. Snce Ω s an open set dstnct of M, we can take (Q ) a Whtney decomposton of Ω. That s the balls Q are parwse dsjont and there exst two constants C 2 > C >, dependng only on the metrc, such that 8 Q

9 . Ω = Q wth Q = C Q and the balls Q have the bounded overlap property; 2. r = r(q ) = 2 d(x, F ) and x s the center of Q ; 3. each ball C 2 Q ntersects F (C 2 = 4C works) and we defne Q = 2C 2 Q. For x Ω, denote I x = { : x Q }. By the bounded overlap property of the balls Q, we have that I x N wth a numercal nteger N. Fxng j I x and usng the propertes of the Q s, we easly see that 3 r r j 3r for all I x. In partcular, Q 7Q j for all I x. Condton (9) s nothng but the bounded overlap property of the Q s and (8) follows from (9) and (0). Observe that the doublng property and the fact that Q F yeld ( f q + f q + AQ f q + AQ f q )dµ ( f q + f q + AQ f q + AQ f q )dµ Q Q nf [M q ( f + f ) + M A,q (f)] q µ(q ) Q α q µ(q ) α q µ(q ). () We now defne the functons b. Let (χ ) be a partton of unty of Ω assocated to the coverng (Q ), such that for all, χ s a Lpschtz functon supported n Q wth χ r. Set b := (f A Q f)χ. It s clear that supp(b ) Q. Let us estmate b W,q (Q ). We have b q dµ = (f AQ (f) q dµ Q Q f q dµ + AQ (f) q dµ Q Q α q µ(q ). We appled () n the last nequalty. Snce ) ( ((f A Q f)χ = χ f AQ f ) + ( f A Q f ) χ, we have b q dµ f AQ (f) q dµ + Q Q r q Q f A Q f q dµ. The frst term s estmated as above for b. Thus Q f AQ (f) q dµ α q µ(q ). 9

10 For the second term, the Poncaré nequalty (P q ) (relatvely to the collecton A) shows that f r q AQ (f) q µ(q ) dµ sup ( f q + f q )dµ Q s µ(sq ) sq α q µ(q ). We used that for all s, sq meets F and () for sq nstead of Q. Therefore (7) s proved. Set g = f b, then t remans to prove (6). Snce the sum s locally fnte on Ω, g s defned almost everywhere on M and g = f on F. Observe that g s a locally ntegrable functon on M. Ths follows from the fact that b = f g L q here (for the homogeneous case, one can easly prove that b L loc ). Note that χ = Ω and χ = Ω. We then have g = f b ( [ = f χ f AQ f ]) (f A Q (f)) χ = F ( f) + χ A Q f A Q (f) χ f Ω. (2) The defnton of F and the Lebesgue dfferentaton theorem yeld F ( f + f ) α µ a.e. We deduce that (wth an nterpolaton nequalty) for r = θ p : F ( f + f ) L r F ( f + f ) θ L p F ( f + f ) ( θ) L f p/r W,p α p/r. We control the second term n (2) usng the off-dagonal decays of A: (4). We recall that Q = 2C 2 Q. We deduce that AQ f L µ(q r (Q ) ) /r nf M q ( f + f ) Q αµ(q ) /r. (3) The last nequalty s due to the fact that Q F. Then the bounded overlap property of the coverng (Q ) gves us χ (x) A Q f L r ( AQ f r L r (Q ) ( α r µ(q ) α(µ(ω)) /r. ) /r ) /r 0

11 We clam that a smlar estmate holds for h = [ AQ (f) f ] χ : we have h L r α(µ(ω)) /r. To prove ths, we fx a pont x Ω and let Q j be a Whtney ball contanng x. For all I x as r Q r Qj, we have A Q (f) A Qj (f) r j µ(q j ) /r α. (4) L r (Q ) Indeed, snce Q 7Q j, ths s a drect consequence of the assumed off-dagonal decays and the fact that 0Q F. Usng χ (x) = 0, we deduce that h L r (Q j ) AQ (f) A Qj (f) r j Nαµ(Q j ) /r αµ(q j ) /r. (5) I x L r (Q j ) Usng agan the bounded overlap property of the (Q ) s, t follows that Hence h L r α(µ(ω)) /r. g L r (Ω) α(µ(ω)) /r. Then (8) and the L r estmate of g on F yeld g L r f p/r W α p/r. Let us now,p estmate g L r. We have g = f F + A Q (f)χ. Snce f F α, stll need to estmate A Q (f)χ L r. Note that as n (3), we smlarly have for every AQ (f) L αµ(q r (Q ) ) /r. (6) As above, ths last nequalty yelds (thanks to the bounded overlap property of the (Q ) ) g L r (Ω) α(µ(ω)) /r. Fnally, (8) and the L r estmate of g on F yeld g L r f p/r W α p/r. Therefore we,p proved that g belongs to W,r wth the desred boundedness. Remark 3.5 Note that n ths decomposton, Ω corresponds to a sngular dstrbuton, supported n Ω. In the prevous proof, we consdered that the dstrbuton Ω corresponds to a functon, vanshng almost everywhere. The estmate (5) shows that h (consdered as an L loc-functon) satsfes the good property. We also have to check that h can be consdered as an L loc-functon. Ths s due to the followng fact [ ] A Qj (f)χ j f χ = 0,j n the dstrbutonal sense. Ths equalty shows that when we are close to supp( χ ) = Ω, the correspondng operator A Qj tends to the dentty operator, due to Poncaré nequalty. We do not detal ths techncal problem and refer to [4]. Remark 3.6 In the case where the operator A Q s the mean-operator over the ball Q, the assumpton M A,q = M q s contnuous from W,p to L p, s always satsfed. The Poncaré nequalty (P q ) corresponds to the classcal one (n fact t s weaker snce that n the classcal one t appears only the L q (Q) norm of the gradent of the functon). Moreover L q L off-dagonal estmates hold obvously. Thus, we regan the well-known Calderón-Zygmund decomposton n Sobolev spaces.

12 3.2 Applcaton to real Interpolaton of Sobolev spaces. As descrbed n [], such a Calderón-Zygmund decomposton n Sobolev spaces s suffcent to obtan a real nterpolaton result for Sobolev spaces. Theorem 3.7 Let M be a complete Remannan manfold of nfnte measure satsfyng (D) and admttng a Poncaré nequalty (P q ) for some q [, ) relatvely to the collecton A. Assume that A satsfes L q L r off-dagonal estmates for an r (q, ]. Then for s p < r wth p > q, the space W,p s a real nterpolaton space between W,s and W,r. More precsely W,p = (W,s, W,r ) θ,p where θ (0, ) such that p := θ + θ s r < q. We do not detal the proof and refer the reader to [] for the lnk between such a Calderón-Zygmund decomposton and nterpolaton results. We brefly explan the man steps of the proof. Proof : It s suffcent to prove that there exsts C > 0 such that for every f W,p and t > 0, K(f, t, W,s, W,r ) ( t r r s [ f q + f q ] /q (t rs r s ) + t [ t rs r s ] ) /r (M( f + f ) q ) r/q (u)du. (7) We consder the prevous Calderón-Zygmund decomposton for f wth α = α(t) = [M q ( f + f ) + M A,q (f)] q q (t rs r s ). We wrte f = b + g = b + g where (b ), g satsfy the propertes of Theorem 3.3. From the bounded overlap property of the B s, t follows that b s W,s N b s W,s α s (t) µ(b ) α s (t)µ(ω t ), wth Ω t = B. For g, we have as n [], proof of Theorem 4.2, p.5 ( g r + g r ) dµ = ( f r + f r ) dµ F t F t t rs r s (M( f + f ) q ) r q (u)du + t rs r s ( f q + f q ) r q (t rs r s ) 2

13 where F t s the complement of Ω t. For the Sobolev norm of g n Ω, we use the estmate of the Calderón-Zygmund decomposton. Moreover, snce (Mf) f and (f + g) f + g (c.f [7],[8]) and thanks to the (L q L r ) off-dagonal assumpton on A, we have ) α(t) ( f q rs q (t r s ) + f q rs q (t r s ). The choce of α(t) mples µ(ω t ) t rs r s (c.f [7],[8]). Fnally (7) follows from the fact that K(f, t, W,s, W,r ) b W,s + t g W,r and the good estmates of b W,s and g W,r. Remark 3.8 As explaned n [0, ], to nterpolate the non-homogeneous Sobolev spaces, t s suffcent to assume local doublng (D loc ) and local Poncaré nequalty (P qloc ) relatvely to A. In these assumptons, we restrct to balls Q of radus suffcently small. We now gve an homogeneous verson of all these results and then gve applcatons. 3.3 Homogeneous verson We begn recallng the defnton of homogeneous Sobolev spaces on a manfold.. E,p Let M be a C Remannan manfold of dmenson n. For p, we defne to be the vector space of dstrbutons ϕ wth ϕ L p, where ϕ s the dstrbutonal. gradent of ϕ. We equp E,p wth the sem-norm The homogeneous Sobolev space ϕ. E,p = ϕ L p.. W,p s then the quotent space E. p/r. Remark 3.9.For all ϕ E.,p, ϕ W. = ϕ,p Lp, where ϕ denotes the class of ϕ.. 2. The space W,p s a Banach space (see [27]). We then have all the homogeneous verson of our results. We only state them, ther proofs beng the same as n the non-homogeneous case wth few modfcatons due to the homogeneous norm. Let A := (A Q ) Q be a collecton of operators (actng from Ẇ,p to Ẇ,p loc ) ndexed by the balls of the manfold. We defne analogously new homogeneous maxmal operator assocated to ths collecton: for s p and all functons f Ẇ,p Ṁ A,s (f)(x) := sup Q; Q x µ(q) /s A Q(f) L s (Q). The assumptons that we need on the collecton A are then the followng: 3

14 Defnton 3.0 ) We say that for q [, ], the manfold M satsfes an homogeneous Poncaré nequalty ( P q ) relatvely to the collecton A f there s a constant C such that for every ball Q (of radus r Q ) and for all functons f Ẇ,p ; p q: ( /q ( /q f A Q (f) dµ) q Cr Q sup f dµ) q. s Q 2) We say that the collecton A satsfes L q L r homogeneous off-dagonal estmates f a. there are constants C > 0 and N N such that for all equvalent balls Q, Q (.e. Q Q NQ; N N ) and all functons f Ẇ,p ; p q, we have µ(q) A Q(f) A /r Q (f) L r (NQ) C r Q nf M q ( f ) NQ b. and for every ball Q µ(q) A Q(f) /r L r (Q) C nf M q ( f ). (8) Q Then, we get the homogeneous verson of the Calderón-Zygmund decomposton: Theorem 3. Let M be a complete Remannan manfold satsfyng (D) and of nfnte measure. Consder a collecton A = (A Q ) Q of operators defned on M. Assume that M satsfes the Poncaré nequalty ( P q ) relatvely to the collecton A for some q [, ) and that A satsfes L q L r homogeneous off-dagonal estmates for an r (q, ]. Let f Ẇ,p and α > 0. Then one can fnd a collecton of balls (Q ), functons g Ẇ,r and b Ẇ,q wth the followng propertes f = g + b (9) g Ẇ,r f p/r α p/r, g r dµ α r µ( Ẇ,p Q ) (20) Q sq supp (b ) Q, b Ẇ,q αµ(q ) /q (2) µ(q ) Cα p f p dµ (22) Q N. (23) Ths decomposton wll gve us the followng homogeneous nterpolaton result: Theorem 3.2 Let M be a complete Remannan manfold of nfnte measure satsfyng (D) and admttng a Poncaré nequalty ( P q ) for some q [, ) relatvely to the collecton A. Assume that A satsfes L q L r homogeneous off-dagonal estmates for an r (q, ]. Then for s p < r wth p > q, the space Ẇ,p s a real nterpolaton space between Ẇ,s and Ẇ,r. More precsely where θ (0, ) such that Ẇ,p = (Ẇ,s, Ẇ,r ) θ,p p := θ + θ s r < q. 4

15 4 Pseudo-Poncaré nequaltes and Applcatons 4. The partcular case of Pseudo-Poncaré Inequaltes Thanks to [2, 3], we know that under (D), a Poncaré nequalty (P q ) guarantees the assumptons of Theorem 3.3 when A Q s the mean-operator over the ball Q. Thus t permts to prove a Calderón-Zygmund decomposton for Sobolev functons. The am of ths subsecton s to show, usng a partcular choce of operators A Q, that our assumptons are weaker than the classcal Poncaré nequalty used n the already known decomposton. Let be the postve Laplace-Beltram operator and let us set A Q := e r2 Q for each ball Q of radus r Q. In all ths secton, we work wth these operators. In order to obtan a Calderón-Zygmund decomposton as n Theorem 3.3, we need to put some assumptons on (A Q ) Q as those n Secton 3. Accordng to ths choce of operators, we defne what are Pseudo-Poncaré nequaltes. Defnton 4. (Pseudo-Poncaré nequalty on M) We say that a complete Remannan manfold M admts a pseudo-poncaré nequalty ( P q ) for some q [, ) f there exsts a constant C > 0 such that, for every functon f C0 and every ball Q of M of radus r > 0, we have ( Q f e r2 f q dµ) /q Cr sup s ( sq f q dµ) /q. ( P q ) Pseudo-Poncaré nequaltes corresponds to what we called Poncaré nequalty relatvely to ths collecton A (the homogeneous verson, we can also consder the non-homogeneous one). We begn showng that pseudo-poncaré nequaltes are mpled by the classcal Poncaré nequaltes. We denote q 0 := nf{q [, ); (P q ) holds }. (q 0 ) Proposton 4.2 Let M be a complete manfold satsfyng (D) and admttng a Poncaré nequalty (P q ) for some q <.. If q 0 < 2 then the pseudo-poncaré nequalty ( P q ) holds. 2. If q 0 2, we moreover assume (DUE). Then ( P q ) also holds. Before provng ths proposton, we gve the followng coverng Lemma. Lemma 4.3 Let M be a complete manfold satsfyng (D). Let Q a ball of radus r Q. Then there exsts a bounded coverng (Q j ) j of Q wth balls of radus t /2 for 0 < t r 2 Q. Moreover, for s, the collecton (sq j ) j s a s-coverng of sq, that s : sup {j, x sq j } s d, x sq where d s the homogeneous dmenson of the manfold. 5

16 Proof : We choose ( Q(x j, t /2 /3) ) a maxmal collecton of dsjont balls n Q. Then we j set Q j = Q(x j, t /2 ), whch s a coverng of Q. Fx x sq and denote J x := {j, x sq j }. Take j 0 J x (f J x otherwse, there s nothng to prove). By (D), we have ( ) ( J x ) µ (sq j0 ) ( J x ) s d µ 3 Q j 0 s ( ) d µ 3 Q j j J x ( ) s d µ j Jx 3 Q j s d µ ( Q(x, 2st /2 ) ) s d µ (sq j0 ), where we used the fact that the balls Q 3 j are dsjont and have equvalent measure when the ndex j J x. Proof of Proposton 4.2 Consder a ball Q of radus r > 0. We deal wth the semgroup and wrte the oscllaton as follows r 2 f e r2 d r 2 f = dt e t fdt = e t fdt. 0 0 Now we apply arguments used n [7], Lemma 3.2. Usng the completeness of the manfold, we have ( r 2 q /q r 2 ( e t fdt dµ) e t f /q dµ) q dt µ(q) µ(q) Q 0 0 r 2 0 ( µ(q) Q j Q j e t (f f Qj ) q dµ ) /q dt, where (Q j ) j s a bounded coverng of Q wth balls of radus t /2 as n Lemma 4.3. Fx t (0, r 2 ) and denote by C k (Q j ) := 2 k+ Q j \ 2 k Q j for k and C 0 (Q j ) = 2Q j. 6

17 Then, argung as n Lemma 3.2 n [7] e t (f f Qj ) q dµ j Q j q t q e cd2 (x,y)/t j Q j M µ(q(y, t)) (f(y) f Q j )dµ(y) dµ(x) t q (µ(2 k+ Q j )) q e cqd2 (x,y)/t j,k;k 0 Q j C k (Q j ) µ(q(y, t)) f(y) f q Q j q dµ(y)dµ(x) ( ) f(y) t q (µ(2 k+ Q j )) q e cqd2 (x,y)/t fqj q dµ(x) j,k;k C k (Q j ) {x; d(x,y) 2 k t} µ(q(y, t)) dµ(y) q + ( ) t q µ(q j j ) (µ(2q j)) q dµ(x) f(y) f q Qj q dµ(y) 2Q j Q j t q e cq4k 2 kdq f(y) fqj q dµ(y) j k C k (Q j ) + j j + j t q 2Q j f(y) fqj q dµ(y) t q k e cq4k 2 kdq 2 k+ Q j f(y) f2 k+ Q j q dµ(y) + t q 2Q j f(y) fqj q dµ(y) k+ l= µ(2 k+ Q j ) µ(2 l Q j ) f 2 l Q j f 2 l Q j j t q k k+ e cq4k 2 Mk t q/2 l= 2 l Q j f q dµ + j t q t q/2 2Q j f q dµ. We used (2), (P q ), that for y 2Q j, µ(q(y, t)) µ(q j ) and for y C k (Q j ), k, µ(q(y, C 2kd t)) µ(2 k+ Q j. We also used that for s, t > 0, ) e cd2 (x,y)/s dµ(x) Ce ct/s µ(q(y, s)) {x; d(x,y) t} thanks to (D) (see Lemma 2. n [24]). Usng that (2 l Q j ) j s a 2 l -bounded coverng of 2 l Q, we deduce that f q dµ 2 ld f q dµ 4 ld µ(q) sup f q dµ, 2 l Q j 2 l Q s sq j where d s the homogeneous dmenson of the doublng manfold. Thus, t follows that ( r 2 q /q [ ] r 2 ( /q e t (f)dt dµ) t /2 dt ( f )dµ) q, µ(q) Q 0 0 sup s sq whch ends the proof. 7

18 Before we prove off-dagonal estmates under the classcal Poncaré nequalty, let us recall the followng result : Proposton 4.4 ([6]) Let M be a complete Remannan manfold satsfyng (D) and (P 2 ). Then there exsts p 0 > 2 such that the Resz transform R := ( ) 2 s L p bounded for < p < p 0. We now let and { p 0 := sup p (2, ); ( ) 2 } s L p bounded (p 0 ) s 0 := sup {s (, ]; (G s ) holds }. (s 0 ) Remark 4.5 Note that the doublng property (D) and (DUE) mply for p (, 2], the L p boundedness of 2 whch mples (G p ) (see Subsecton 2.3) and that s 0 p 0 > 2. For the second off-dagonal condton (4), we obtan : Proposton 4.6 Let M be a complete manfold. Assume that M satsfes (D) and admts a classcal Poncaré nequalty (P q ) for some q [, ) as n Defnton 2.4. Consder the followng estmate M A,r (f) M q ( f + f ). (24). If q 0 < 2, then (24) holds for all r (q, s 0 ). 2. If q 0 2, assume moreover (DUE) and that s 0 > q. Then (24) holds for all r (q, s 0 ). Consequently, (4) holds for all r (q, s 0 ). Proof : It s suffcent to prove the followng nequaltes ( /r e r2 f dµ) r CM q ( f )(x) (25) Q Q and ( /r e r2 f dµ) r CM q ( f )(x) (26) for every x M and every ball Q contanng x. We do not detal the proof as t uses analogous argument as n [7], subsecton 3., Lemma 3.2 and the end of ths subsecton. For example, (26) s essentally nequalty (3.2) n secton 3 of [7] where q 0 = 2. We just menton that for (25), we use the L r contractvty of the heat semgroup, (D) and (DUE). For (26), we moreover need the followng L r -Gaffney estmates for e t wth r (q 0, s 0 ). We say that ( e t ) t>0 satsfes the L p Gaffney estmate f there exsts C, α > 0 such that for all t > 0, E, F closed subsets of M and f supported n E t e t f L p (F ) Ce αd(e,f )2 /t f L p (E). (Ga p ) In the case where q 0 2, nterpolatng the already known (Ga 2 ) wth (G s ) for every 2 < s < s 0, we get the (Ga p ) for 2 < p < s 0. When q 0 < 2, snce n ths case (G s ) holds for all < s < 2 and 2 < s < s 0, nterpolatng agan (G s ) and (Ga 2 ), we obtan the (Ga p ) for all < p < s 0. It remans to check (3). 8

19 Proposton 4.7 Let M be a complete manfold satsfyng (D) and admttng a classcal Poncaré nequalty (P q ) for some q <. Then. If q 0 < 2, for r > q, the collecton A satsfes (L q L r ) off-dagonal estmates (3). 2. If q 0 2, the same result holds under the addtonal assumpton (DUE). Proof : Take Q 0, Q two equvalent balls, let us say Q 0 Q 0Q 0 wth radus r 0 (resp. r ). We choosed a numercal factor 0 just for convenence. We have to prove that ( e r2 0 f e r2 f r /r dµ) r 0 nf M q ( f + f ). (27) µ(q 0 ) 0Q 0Q 0 0 Ths s a consequence of ( e r2 0 f e 400r2 0 f r /r dµ) r 0 nf M q ( f + f ) (28) µ(q 0 ) 0Q 0Q 0 0 and ( e 400r2 0 f e r2 f r /r dµ) r 0 nf M q ( f + f ). (29) µ(q 0 ) 0Q 0Q 0 0 We use that e r2 0 f e 400r2 0 f = e r2 0 [ e 399r2 0 ] (f) and e 400r2 0 f e r2 f = e r2 [ e (20r 0) 2 r 2 ) ] (f). We only deal wth (28), we do the same for (29). From (D) and (DUE), we know that (UE) holds and so we have very fast decays (L L ) for the semgroup, whch permts to gan ntegrablty from L q to L r. It follows ( e r2 0 f e 400r2 0 f r dµ µ(q 0 ) 0Q 0 ( e γ4j µ(q j 0 0 ) C j (Q 0 ) ) /r f e 399r2 0 f /q dµ) q, where we make appear the dyadc coronas C j (Q 0 ) (see agan [7], Lemma 3.2 and the end of subsecton 3.). Then we use (D) and (P q ). For each j, we choose a bounded coverng 9

20 (Q j ) of 2 j+ Q 0 wth balls of radus 399r 0 and obtan f e 399r2 0 f q dµ f e 399r2 0 f q dµ µ(q 0 ) C j (Q 0 ) µ(q 0 ) Q j f e 399r2 0 f q dµ µ(q 0 ) Q j ( ) r q µ(q 0 ) 0µ(Q j ) sup f q dµ s µ(q 0 ) µ(q 0 ) r q 02 dj µ(2j+ Q 0 ) µ(q 0 ) sq j ) r0µ(q q j ) sup 2 ( dj f q dµ s s2 j+ Q 0 r0µ(q q j )2dj nf M ( f q ) Q 0 ( ) q nf M q ( f ) Q 0 ) q. r q 02 2dj (nf Q 0 M q ( f ) We appled (P q ) n the thrd nequalty. In the fourth nequalty, we used that sq j 2 j+ sq 0 and thanks to (D), µ(2 j+ sq 0 ) µ(sq j )2jd. Then we appled the bounded overlap property n the sxth one. Summng n j, we show the desred nequalty (28). Smlarly we prove (29), whch completes the proof of (27). We get the followng corollary: Corollary 4.8 Assume that M s complete, satsfes (D) and admts a classcal Poncaré nequalty (P q ) for some q [, ). In the case where q 0 2, we moreover assume (DUE) and s 0 > q. Then the assumptons of Theorem 3.3 and 3.7 hold. We have pseudo-poncaré nequalty ( P q ) and A satsfes L q L r off-dagonal estmates for r (q, s 0 ). Concluson : When q < 2, the assumptons of Theorem 3.3 (accordng to ths partcular choce of A) are weaker than the Poncaré nequalty and are suffcent to get the Calderón- Zygmund decomposton. We also have the homogeneous verson: Corollary 4.9 Assume that M s complete, satsfes (D) and admts a classcal Poncaré nequalty (P q ) for some q <. In the case where q 0 2, we moreover assume (DUE). Let A := (A Q ) Q wth A Q := e r2 Q. Then the assumptons of Theorems 3. and 3.2 holds. We have pseudo-poncaré nequalty ( P q ), A satsfes homogeneous L q L r offdagonal estmates for r (q, s 0 ). 20

21 4.2 Applcaton to Reverse Resz transform nequaltes. We refer the reader to [6, 7] for the study of the so-called (RR p ) nequaltes : /2 f L p f L p. (RR p ) We know that (RR 2 ) s always satsfed and that (D) and (DUE) mples (RR p ) for all p (2, ). For the exponents lower than 2, P. Auscher and T. Coulhon obtaned the followng result ([6]) : Theorem 4.0 Let M be a complete non-compact doublng Remannan manfold. Moreover assume that the classcal Poncaré nequalty (P q ) holds for some q (, 2). Then for all p (q, 2), (RR p ) s satsfed. Ths result s based on a Calderón-Zygmund decomposton for Sobolev functons. Usng our new assumptons, we also obtan the followng mprovement : Theorem 4. Assume that M s complete, satsfes (D) and admts a pseudo-poncaré nequalty ( P q ) for some q (, 2). If n addton, the collecton A satsfes L q L 2 off-dagonal estmates, then (RR p ) holds for all p (q, 2). Remark 4.2 Corollary 4.8 shows that these new assumptons are weaker than the Poncaré nequalty (P q ). We do not prove ths result and refer the reader to [6]. The proof s exactly the same as t reles on the Calderón-Zygmund decomposton. Remark 4.3 We refer the reader to other works of the authors [2, 5]. In [2], the assumpton (RR p ) plays an mportant role n order to prove some maxmal nequaltes n dual Sobolev spaces W,p, whch do not requre Poncaré nequaltes. So t mght be mportant to know how to prove (RR p ) wthout Poncaré nequalty. 4.3 Applcaton to Gaglardo-Nrenberg nequaltes. We devote ths subsecton to the study of Gaglardo-Nrenberg nequaltes. We refer the reader to [2] for a recent work on ths subject. Defnton 4.4 We ntroduce the Besov space. For α < 0, we set B α, the set of all measurable functons f such that f B α, := sup t α 2 e t f L <. t>0 We have the followng equvalence (Lemma 2. n [2]) : f B α, sup t α 2 e t (f e t f) L. t>0 2

22 Then, the so-called Gaglardo-Nrenberg nequaltes are : f l f θ p f θ θ θ B, (30) where θ = p for some p, l [, ). l We frst recall one of the man results of [2]: Theorem 4.5 Let M be a complete non-compact Remannan manfold satsfyng (D) and (P q ) for some q <. Moreover, assume that M satsfes the global pseudo- Poncaré nequaltes (P q) and (P ). Then (30) holds for all q p < l <. Here, the global pseudo-poncaré nequalty (P q) for some q [, ] corresponds to f e t f L q Ct 2 f L q. (P q) Ths result requres global pseudo-poncaré nequaltes and some Poncaré nequaltes wth respect to balls. These two knds of nequaltes are qute dfferent as they deal wth oscllatons wth respect to the semgroup (for the pseudo-poncaré nequaltes) and to the mean value operators (for the Poncaré nequaltes). We saw n the prevous subsecton, that Poncaré nequalty mples pseudo-poncaré nequalty. That s why, we are lookng for assumptons requrng only the Poncaré nequalty, gettng around the assumed global pseudo-poncaré nequaltes. We begn frst showng that pseudo-poncaré nequaltes related to balls yeld global pseudo-poncaré nequaltes. Proposton 4.6 Let M be a complete Remannan manfold satsfyng (D) and admttng a pseudo-poncaré nequalty ( P q ) for some q <. Then the global pseudo- Poncaré nequalty (P q) holds. Proof : Let t > 0. Pck a countable set {x j } j J M, such that M = Q(x j, t) := j J Q j and for all x M, x does not belong to more than N balls Q j. Then j J f e t f q q f e t f q dµ j Q j t q 2 f q dµ j Q j N t q 2 f q dµ. Remark 4.7 It s easy to see that the global pseudo-poncaré nequalty (P ) s satsfed under (D) and (DUE) (see for nstance [2], p.499). M 22

23 Usng Propostons 4.6, 4.2 and Theorem 4.5, we get the followng mprovement verson of Theorem.2 n [2] : Theorem 4.8 Let M be a complete Remannan manfold satsfyng (D) and admttng a Poncaré nequalty (P q ) for some q <. If q 0 2, we moreover assume (DUE). Then (30) holds for all q p < l <. Usng our new assumptons, we get also the followng Gaglardo-Nrenberg theorem: Theorem 4.9 Assume that M satsfes the hypotheses of Theorem 3.2 wth A Q = e r2 Q and that r =. Moreover, we assume (DUE). Then (30) holds for all q p < l <. Proof : The proof s analogous to that of Theorems. and.2 n [2]. We use our homogeneous nterpolaton result of Theorem 3.2. Also we need our non-homogeneous nterpolaton result of Theorem 3.7. It holds thanks to (25) whch s true under (D) and (DUE). Moreover, (P q) s satsfed and (P ) holds thanks to (D) and (DUE). As a Corollary, we obtan Theorem 4.20 Consder a complete Remannan manfold M satsfyng (D), (P q ) for some q < and assume that there exsts C > 0 such that for every x, y M and t > 0 C x p t (x, y). (G) tµ(b(y, t)) ((G) s equvalent to the assumpton (G ).) In the case where q 0 > 2, we moreover assume (DUE). Then nequalty (30) holds for all q p < l <. Proof : In the case where q 2, ths result s already n [2]. For q 0 2, we are under the hypotheses of Theorem 4.9 thanks to subsecton 4. and snce (G) mples that r =. References [] L. Ambroso, M. Mranda Jr, and D. Pallara. Specal functons of bounded varaton n doublng metrc measure spaces. Calculus of varatons : topcs from the mathematcal hertage of E. De Gorg, Quad. Mat., Dept. Math, Seconda Unv. Napol, Caserta, 4, 45, [2] P. Auscher. On L p estmates for square roots of second order ellptc operators on R n. Publ. Mat. 48, 59 86, [3] P. Auscher. On necessary and suffcent condtons for L p estmates of Resz transforms assocated to ellptc operators on R n and related estmates. Memors of Amer. Math. Soc. 86, (87), [4] P. Auscher. On the Calderón-Zygmund lemma for Sobolev functons: 23

24 [5] P. Auscher and B. Ben Al. Maxmal nequaltes and Resz transform estmates on L p spaces for Schrödnger operators wth nonnegatve potentals. Ann. Inst. Fourer, 57, (6), , [6] P. Auscher and T. Coulhon. Resz transform on manfolds and Poncaré nequaltes. Ann. Sc. Nor. Sup. Psa (5), IV, 3, , [7] P. Auscher, T. Coulhon, X. T. Duong and S. Hofmann. Resz transform on manfolds and heat kernel regularty, Ann. Sc. Ecole Norm. Sup. 37, 9-957, [8] P. Auscher and J.M. Martell. Weghted norm nequaltes, off-dagonal estmates and ellptc operators. Part I : General operator theory and weghts. Adv. n Math. 22, , [9] N. Badr. Ph.D Thess. Unversté Pars-sud, [0] N. Badr. Real nterpolaton of Sobolev spaces, Math. Scand. 3, (4), , [] N. Badr. Real nterpolaton of Sobolev spaces assocated to a weght, Pot. Anal. 3, (4), , [2] N. Badr. Gaglardo-Nrenberg nequaltes on manfolds. J.M.A.A, 349, (2), , [3] N. Badr and E. Russ. Interpolaton of Sobolev Spaces, Lttlewood-Paley nequaltes and Resz transforms on graphs, Pub. Mat., 53, (2), , [4] N. Badr and B. Ben Al. L p boundedness of Resz tranform related to Schrödnger operators on a manfold, Ann. Sc. Nor. Sup. Psa. VIII, (4), , [5] N. Badr and F. Berncot. Abstract Hardy-Sobolev spaces and nterpolaton, to appear n J. Funct. Anal, avalable at [6] B. Ben Al. Ph.D Thess. Unversté Pars-sud, [7] C. Bennett and R. Sharpley. Interpolaton of operators. Academc Press, 988. [8] J. Bergh and J. Löfström. Interpolatons spaces, An ntroducton. Sprnger (Berln), 976. [9] F. Berncot and J. Zhao. Abstract Hardy spaces, J. Funct. Anal. 255, (7), , [20] F. Berncot. Real nterpolaton of abstract Hardy spaces and applcatons to the blnear theory., Math. Z. 265, , 200. [2] F. Berncot. Maxmal nequaltes for dual Sobolev spaces W,p and applcatons to nterpolaton, Mathematcal Research Letters 6, (5), , [22] R. Cofman and G. Wess. Analyse harmonque sur certans espaces homogènes. Lecture notes n Math., Sprnger,

25 [23] T. Coulhon and X.T. Duong. Resz transforms for p 2. Trans. Amer. Math. Soc. 35, (2), 5 69, 999. [24] T. Coulhon and X.T. Duong. Maxmal regularty and kernel bounds: observatons on a theorem by Heber and Prüss. Adv. Dfferental Equatons 5, pages , [25] E. B. Daves. Non-Gaussan aspects of heat kernel behavor. J. London Math. Soc., 997. [26] X.T. Duong, L. Yan. Dualty of Hardy and BMO spaces assocated wth operators wth heat kernel bounds. J. Amer. Math. Soc. 8, (4), , [27] V. Gol dshten and M. Troyanov. Axomatc Theory of Sobolev Spaces. Expo. Mathe., 9, , 200. [28] A. Grgor yan. Gaussan upper bounds for the heat kernel on arbtrary manfolds. J. Dff. Geom. 45, 33 52, 997. [29] P. Hajlasz and P. Koskela. Sobolev met Poncaré. Mem. Amer. Math. Soc. 45, (688), 0, [30] S. Keth and X. Zhong. The Poncaré nequalty s an open ended condton. Ann. of Maths. 67, (2), , [3] P. L and S.T. Yau, On the parabolc kernel of the Schrödnger operator. Acta Math. 56, 53 20, 986. [32] J.M. Martell. Sharp maxmal functons assocated wth approxmatons of the dentty n spaces of homogeneous type and applcatons. Studa Math. 6, 3 45, [33] L. Saloff-Coste. A note on Poncaré, Sobolev and Harnack nequaltes. Duke J. Math. 65, 27 38, 992. [34] E.M. Sten, Sngular ntegrals and dfferentablty propertes of functons, Prnceton Unv. Press,