L p Boundedness of Riesz transform related to Schrödinger operators on a manifold

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1 L p oundedness of Resz transform related to Schrödnger operators on a manfold Nadne adr esma en Al Unversté Lyon Unversté Pars-Sud Aprl 7, 009 Abstract We establsh varous L p estmates for the Schrödnger operator + V on Remannan manfolds satsfyng the doublng property and a Poncaré nequalty, where s the Laplace-eltram operator and V belongs to a reverse Hölder class. At the end of ths paper we apply our result on Le groups wth polynomal growth. Contents Introducton Prelmnares 6. The doublng property and Poncaré nequalty Reverse Hölder classes Homogeneous Sobolev spaces assocated to a weght V Defnton of Schrödnger operator 9 4 Prncpal tools 4. An mproved Fefferman-Phong nequalty Calderón-Zygmund decomposton Estmates for subharmonc functons axmal nequaltes 9 6 Complex nterpolaton Insttut Camlle Jordan, Unversté Claude ernard, Lyon, UR du CNRS 508, 43 boulevard du novembre 98, F-696 Vlleurbanne cedex. Emal: badr@math.unv-lyon.fr Unversté de Pars-Sud, UR du CNRS 868, F-9405 Orsay cedex. Emal:besmath@yahoo.fr

2 7 Proof of Theorem Proof of pont. of Theorem Estmates for weak solutons A reducton Proof of pont. of Theorem Case of Le groups 34 Introducton The man goal of ths paper s to establsh the L p boundedness for the Resz transforms ( + V ), V ( + V ) and related nequaltes on certan classes of Remannan manfolds. Here, V s a non-negatve, locally ntegrable functon on. For the Eucldan case, ths subject was studed by many authors under dfferent condtons on V. We menton the works of Helffer-Nourrgat [35], Gubourg [3], Shen [5], Skora [5], Ouhabaz [47] and others. Recently, Auscher-en Al [3] proved L p maxmal nequaltes for these operators under less restrctve assumptons. They assumed that V belongs to some reverse Hölder class RH q (for a defnton, see secton ). A natural step further s to extend the above results to the case of Remannan manfolds. For Remannan manfolds, the L p boundedness of the Resz transform of + V was dscussed by many authors. We menton eyer [45], akry [9] and Yosda [59]. The most general answer was gven by Skora [5]. Let satsfyng the doublng property (D) and assume that the heat kernel verfes p t (x,.) for all C µ((x, t)) x and t > 0. Under these hypotheses, Skora proved that f V L loc (), V 0, then the Resz transforms of + V are L p bounded for < p and of weak type (, ). L [4] obtaned boundedness results on Nlpotent Le groups under the restrcton V RH q and q D, D beng the dmenson at nfnty of G (see [3]). Followng the method of [3], we obtan new results for p > on complete Remannan manfolds satsfyng the doublng property (D), a Poncaré nequalty (P ) and takng V n some RH q. For manfolds of polynomal type we obtan addtonal results. Ths ncludes Nlpotent Le groups. Let us summarze the content of ths paper. Let be a complete Remannan manfold satsfyng the doublng property (D) and admttng a Poncaré nequalty (P ). Frst we obtan the range of p for the followng maxmal nequalty vald for u C0 (): u p + V u p ( + V )u p. () Here and after, we use u v to say that there exsts a constant C such that u Cv. The startng step s the followng L nequalty for u C 0 (), u + V u 3 ( + V )u () whch holds for any non-negatve potental V L loc (). Ths allows us to defne + V as an operator on L () wth doman D ( ) D (V ).

3 For larger range of p, we assume that V L p loc () and + V s a pror defned on C0. The valdty of () can be obtaned f one mposes for the potental V to be more regular: Theorem.. Let be a complete Remannan manfold satsfyng (D) and (P ). Consder V RH q for some < q. Then there s ɛ > 0 dependng only on V such that () holds for < p < q + ɛ. Ths new result for Remannan manfolds s an extenson of the one of L [4] on Nlpotent Le groups settngs obtaned under the restrcton q D. The second purpose of our work s to establsh some L p estmates for the square root of + V. Notce that we always have the dentty u + V u = ( + V u, u C 0 (). (3) The weak type (, ) nequalty proved by Skora [5] s satsfed under our hypotheses: Interpolatng (3) and (4), we obtan u, + V u, ( + V u. (4) u p + V u p ( + V u p (5) when < p < and u C 0 (). Here, p, s the norm n the Lorentz space L p,. It remans to fnd good assumptons on V and to obtan (5) for some/all < p <. Recall before the followng result Proposton.. ([4]) Let be a complete Remannan manfold satsfyng (D) and (P ). Then there exsts p 0 > such that the Resz transform T = ( ) s L p bounded for < p < p 0. } s L p bounded. We obtan the fol- We now let p 0 = sup lowng theorem. { p ], [; ( ) Theorem.3. Let be a complete Remannan manfold. Let V RH q for some q > and ɛ > 0 such that V RH q+ɛ.. Assume that satsfes (D) and (P ). Then for all u C 0 (), u p ( + V u p for < p < nf(p 0, (q + ɛ)); (6) V u p ( + V u p for < p < (q + ɛ). (7). Assume that s of polynomal type and admts (P ). Suppose that D < p 0, where D s the dmenson at nfnty and that D q < p 0. a. If q < D, then (6) holds for < p < nf(q D + ɛ, p 0), (q D = Dq D q ). b. If q D, then (6) holds for < p < p 0. 3

4 Some remarks concernng ths theorem:. Note that pont s true wthout any addtonal assumpton on the volume growth of balls other than (D). Our assumpton that s of polynomal type n pont whch s stronger than the doublng property (see secton ) s used only to mprove the L p boundedness of ( + V ) when D < q < p 0. We do not need t to prove L p estmates for V ( + V ).. If q > p 0 then we can replace q n pont by any q < p 0 snce V RH q (see Proposton. n secton ). 3. If p 0 D and q D, then (6) holds for < p < p 0 and that s why we assumed D < p 0 n pont.. 4. Fnally the parameter ɛ depends on the self-mprovement of the reverse Hölder condton (see Theorem. n secton ). We establsh also a converse theorem whch s a crucal step n provng Theorem.3. Theorem.4. Let be a complete Remannan manfold satsfyng (D) and (P l ) for some l <. Consder V RH q for some q >. Then and and l < p <. ( + V u l, u l + V u l for every u C 0 () (8) ( + V u p u p + V u p for every u C 0 () (9) Usng the nterpolaton result of [8], we remark that (9) follows drectly from (8) and the the L estmate (3). Remark.5. The estmate (9) always holds n the range p >. Ths follows from the fact that (5) holds for < p and that (5) for p mples (9) for p, where p s the conjugate exponent of p. In the followng corollares we gve examples of manfolds satsfyng our hypotheses and to whch we can apply the theorems above. Corollary.6. Let be a complete n-remannan manfold wth non-negatve Rcc curvature. Then Theorem., pont of Theorem.3 and Theorem.4 hold wth p 0 =. oreover, f satsfes the maxmal volume growth µ() cr n for all balls of radus r > 0 then pont of Theorem.3 also holds. Proof. It suffces to note that n ths case satsfes (D) wth log C d = n, (P ) see Proposton.9 below, that the Resz transform s L p bounded for < p < [9] and that has at most an Eucldean volume growth, that s µ() Cr n for any ball of radus r > 0 Theorem 3.9 n [5]. 4

5 Corollary.7. Let C(N) = R + N be a concal manfold wth compact bass N of dmenson n. Then Theorem., Theorem.4 and Theorem.3 hold wth d = D = n, p 0 = p 0 (λ ) > n where λ s the frst postve egenvalue of the Laplacan on N. Proof. Note that such a manfold s of polynomal type n C r n µ() Cr n for all ball of C(N) of radus r > 0 (Proposton.3, [43]). C(N) admts (P ) [], and even (P ) usng the methods n [30]. For the L p boundedness of the Resz transform t was proved by L [4] that p 0 = when λ n and p 0 = > n n n λ +( n ) when λ < n. Remark.8. Related results for asymptotcally concal manfolds are obtaned n [3] (see the ntroducton and Theorem.5). Under our geometrc assumptons (.e. concal manfolds), loc. ct. and our work are partally complementary. Indeed, the potentals n [3] are requred to have some knd of fast decay at nfnty, whle the Reverse Hölder condton we mposed rules out ths possblty. Our man tools to prove these theorems are: the fact that V belongs to a Reverse Hölder class; an mproved Fefferman-Phong nequalty; a Calderón-Zygmund decomposton; reverse Hölder nequaltes nvolvng the weak soluton of u + V u = 0; complex nterpolaton; the boundedness of the Resz potental when satsfes µ((x, r)) Cr λ for all r > 0. any arguments follow those of [3] wth addtonal techncal problems due to the geometry of the Remannan manfold but those for the Fefferman-Phong nequalty requre some sophstcaton. Ths Fefferman-Phong nequalty wth respect to balls s new even n the Eucldean case. In [3], ths nequalty was proved wth respect to cubes nstead of balls whch greatly smplfes the proof. We end ths ntroducton wth a plan of the paper. In secton, we recall the defntons of the doublng property, Poncaré nequalty, reverse Hölder classes and homogeneous Sobolev spaces assocated to a potental V. Secton 3 s devoted to defne the Schrödnger operator. In secton 4 we gve the prncpal tools to prove the theorems mentoned above. We establsh an mproved Fefferman-Phong nequalty, make a Calderón-Zygmund decomposton, gve estmates for postve subharmonc functons. We prove Theorem. n secton 5. We handle the proof of Theorem.3, pont n secton 6. Secton 7 s concerned wth the proof of Theorem.4. In secton 8, we gve dfferent estmates for the weak soluton of u + V u = 0 and complete 5

6 the proof of tem. of Theorem.3. Fnally, n secton 9, we apply our result on Le groups wth polynomal growth. Acknowledgements. The two authors would lke to thank ther Ph.D advsor P. Auscher for proposng ths jont work and for the useful dscussons and advce on the topc of the paper. Prelmnares Let denote a complete non-compact Remannan manfold. We wrte ρ for the geodesc dstance, µ for the Remannan measure on, for the Remannan gradent, for the Laplace-eltram operator, for the length on the tangent space (forgettng the subscrpt x for smplcty) and p for the norm on L p (, µ), p +.. The doublng property and Poncaré nequalty Defnton. (Doublng property). Let be a Remannan manfold. Denote by (x, r) the open ball of center x and radus r > 0. One says that satsfes the doublng property (D) f there exsts a constant C d > 0, such that for all x, r > 0 we have µ((x, r)) C d µ((x, r)). (D) Lemma.. Let be a Remannan manfold satsfyng (D) and let s = log C d. Then for all x, y and θ and µ((x, θr)) Cθ s µ((x, R)) (0) µ((y, R)) C( + We have also the followng lemma: d(x, y) R )s µ((x, R)). () Lemma.3. Let be a Remannan manfold satsfyng (D). Then for x 0, r 0 > 0, we have µ((x, r)) µ((x 0, r 0 )) 4 s ( r r 0 ) s whenever x (x 0, r 0 ) and r r 0. Theorem.4 (axmal theorem). ([8]) Let be a Remannan manfold satsfyng (D). Denote by the uncentered Hardy-Lttlewood maxmal functon over open balls of X defned by f(x) = sup f :x where f E := fdµ := fdµ. Then E µ(e) E. µ({x : f(x) > λ}) C λ f dµ for every λ > 0; 6

7 . f p C p f p, for < p. Defnton.5. A Remannan manfold s of polynomal type f there s c, C > 0 such that c r d µ((x, r)) cr d (LU l ) for all x and r and for all x and r. C r D µ((x, r)) Cr D (LU ) We call d the local dmenson and D the dmenson at nfnty. Note that f s of polynomal type then t satsfes (D) wth s = max(d, D). oreover, for every λ [mn(d, D), max(d, D)], µ((x, r)) cr λ (L λ ) for all x and r > 0. Defnton.6 (Poncaré nequalty). Let be a complete Remannan manfold, l <. We say that admts a Poncaré nequalty (P l ) f there exsts a constant C > 0 such that, for every functon f C0 (), and every ball of of radus r > 0, we have ( f f l l dµ ( Cr f l l dµ. (Pl ) Remark.7. Note that f (P l ) holds for all f C 0, then t holds for all f W p,loc for p l (see [34], [38]). The followng result from Keth-Zhong [38] mproves the exponent n the Poncaré nequalty. Lemma.8. Let (X, d, µ) be a complete metrc-measure space satsfyng (D) and admttng a Poncaré nequalty (P l ), for some < l <. Then there exsts ɛ > 0 such that (X, d, µ) admts (P p ) for every p > l ɛ. Proposton.9. Let be a complete Remannan manfold wth non-negatve Rcc curvature. Then satsfes (D) (wth C d = n ) and admts a Poncaré nequalty (P ). Proof. Indeed f the Rcc curvature s non-negatve that s there exsts a > 0 such that R c a g, a result by Gromov [6] shows that µ((x, r)) n µ((x, r)) for all x, r > 0. Here n means the topologc dmenson. On the other hand, user s nequalty [3] gves us u u dµ c(n)r u dµ. Thus we get (D) and (P ) (see also [48]). 7

8 . Reverse Hölder classes Defnton.0. Let be a Remannan manfold. A weght w s a non-negatve locally ntegrable functon on. The reverse Hölder classes are defned n the followng way: w RH q, < q <, f. wdµ s a doublng measure.. There exsts a constant C such that for every ball ( w q q dµ C wdµ. () The endpont q = s gven by the condton: w RH whenever, wdµ s doublng and for any ball, w(x) C w for µ a.e. x. (3) On R n, the condton wdµ doublng s superfluous. Remannan manfold. It could be the same on a Proposton.. ([57], [8]). RH RH q RH p for < p q.. If w RH q, < q <, then there exsts q < p < such that w RH p. 3. We say that w A p for < p < f there s a constant C such that for every ball ( ) ( ) p wdµ w p dµ C. For p =, w A f there s a constant C such that for every ball wdµ Cw(y) for µ a.e.y. We let A = p< A p. Then A = <q RH q. Proposton.. (see secton n [3], [36]) Let V be a non-negatve measurable functon. Then the followng propertes are equvalent:. V A.. For all r ]0, [, V r RH. r 3. There exsts r ]0, [, V r RH. r We end ths subsecton wth the followng lemma: 8

9 Lemma.3. (Lemma.4 of []) Let G be an open subset of an homogeneous space (X, d, µ) and let F(G) be the set of metrc balls contaned n G. Suppose that for some 0 < q < p and non-negatve f L p loc, there s a constant A > and σ 0 σ 0 such that ( f p dµ ( p A σ 0 f q q dµ : σ 0 F(G). Then for any 0 < r < q and < σ σ < σ 0, there exsts a constant A > such that ( ( f p p dµ A f r r dµ : σ F(G). σ.3 Homogeneous Sobolev spaces assocated to a weght V Defnton.4. ([8]) Let be a Remannan manfold, V A. Consder for p <, the vector space Ẇ p,v of dstrbutons f such that f and V f Lp. It s well known that the elements of Ẇp,V are n Lp loc. We equp Ẇ p,v wth the sem norm f Ẇ = f p + V f p. p,v In fact, ths expresson s a norm snce V A yelds V > 0 µ a.e. For p =, we denote Ẇ,V the space of all Lpschtz functons f on wth V f <. Proposton.5. ([8]) Assume that satsfes (D) and admts a Poncaré nequalty (P s ) for some s < and that V A. Then, for s p, Ẇ p,v s a anach space. Proposton.6. Under the same hypotheses as n Proposton.5, the Sobolev space Ẇ p,v s reflexve for s p <. Proof. The anach space Ẇ p,v s sometrc to a closed subspace of Lp (, R T ) whch s reflexve. The sometry s gven by the lnear operator T : Ẇp,V Lp (, R T ) such that T f = (V f, f) by defnton of the norm of Ẇp,V and Proposton.5. Theorem.7. ([8]) Let be a complete Remannan manfold satsfyng (D). Let V RH q for some < q and assume that admts a Poncaré nequalty (P l ) for some l < q. Then, for p < p < p q, wth p > l, Ẇp,V s a real nterpolaton space between Ẇ p,v and Ẇ p,v. 3 Defnton of Schrödnger operator Let V be a non-negatve, locally ntegrable functon on. Consder the sesqulnear form Q(u, v) = ( u v + V u v)dµ 9

10 wth doman V = D(Q) = W,V = {f L () ; f & V f L ()} equpped wth the norm f V = ( f + f + V f. Clearly Q(.,.) s a postve, symmetrc closed form. It follows that there exsts a unque postve self-adjont operator, whch we call H = + V, such that Hu, v = Q(u, v) u D(H), v V. When V = 0, H = s the Laplace-eltram operator. Note that C 0 () s dense n V (see the Appendx n [8]). The eurlng-deny theory holds on, whch means that ɛ(h +ɛ) s a postvtypreservng contracton on L p () for all p and ɛ > 0. oreover, f V L loc () such that 0 V V and H s the correspondng operator then one has for any ɛ > 0 and for any f L p, p, f 0 0 (H + ɛ) f (H + ɛ) f. It s equvalent to a pontwse comparson of the kernels of resolvents. In partcular, f V s bounded from below by some postve constant ɛ > 0, then H s bounded on L p for p and s domnated by ( + ɛ) (see Ouhabaz [47]). Let V be the closure of C 0 () under the sem-norm f V = ( f + V f. Assume that satsfes (D) and (P ). y Fefferman-Phong nequalty Lemma 4. n secton 4 below, there s a contnuous ncluson V L loc f V s not dentcally 0, whch s assumed from now on, hence, ths s a norm. Let f V. Then, there exsts a unque u V such that u v + V u v = f, v v C0 (). (4) In partcular, u + V u = f holds n the dstrbutonal sense. We can obtan u for a nce f by the next lemma. Lemma 3.. Assume that satsfes (D) and (P ). Consder f C 0 () L (). For ɛ > 0, let u ɛ = (H + ɛ) f D(H). Then (u ɛ ) s a bounded sequence n V whch converges strongly to H f. Proof. The proof s analogous to the proof of Lemma 3. n [3]. Remark 3.. Assume that satsfes (D) and (P ). The contnuty of the ncluson V L loc () has two further consequences. Frst, we have that L comp(), the space of compactly supported L functons on, s contnuously contaned n V L (). Second, (u ɛ ) has a subsequence convergng to u almost everywhere. 0

11 Fnally as H s self-adjont, t has a unque square root whch we denote H. H s defned as the unque maxmal-accretve operator such that H H = H. We have that H s self-adjont wth doman V and for all u C0 (), H u = u + V u. Ths allows us to extend H from V nto L (). If S denotes ths extenson, then we have S S = H where S : L () V s the adjont of S. 4 Prncpal tools We gather n these secton the man tools that we need to prove our results. Some of them are of ndependent nterest. 4. An mproved Fefferman-Phong nequalty Lemma 4.. Let be a complete Remannan manfold satsfyng (D). Let w A and p <. Assume that admts also a Poncaré nequalty (P p ). Then there are constants C > 0 and β dependng only on the A constant of w, p and the constants n (D), (P p ), such that for every ball of radus R > 0 and u Wp,loc ( u p + w u p )dµ Cm β(r p w ) u p dµ (5) R p where m β (x) = x for x and m β (x) = x β for x. Proof. Snce admts a (P p ) Poncaré nequalty, we have u p C dµ u(x) u(y) p dµ(x)dµ(y). R p µ() Ths and lead easly to w u p dµ = w(x) u(x) p dµ(x)dµ(y) µ() ( u p + w u p )dµ [mn(cr p, w)] u p dµ. Now we use that w A. There exsts ε > 0, ndependent of, such that E = {x : w(x) > εw } satsfes µ(e) > µ(). Hence [mn(cr p, w)] mn(cr p, εw ) C mn(r p, w ). Ths proves the desred nequalty when R p w. Assume now R p w >. We say that a ball of radus R s of type f R p w < and of type f not. Take δ, ɛ > 0 such that δ < ɛ <. We consder a maxmal coverng of ( ɛ) by balls ( ) := ((x, δr)) such that the balls are parwse dsjont. y (D) there exsts N ndependent of δ and R such that I N. Snce δ < ɛ, we have for all I. Denote G the unon of all balls of type

12 and G = {x : d(x, G ) ɛδr}. Set Ẽ = ( ɛδ) \ G. Ths tme we consder a maxmal coverng of Ẽ by balls ( ) := ((x, δ R)) such that the balls are parwse dsjont. Therefore wth the same N one has I N. Let G be the unon of all balls of type and G = {x : d(x, G G ) ɛδ R}, Ẽ = ( ɛδ ) \ G. We terate ths process. Note that the G j s are parwse dsjont (from δ < ɛ). We clam then that µ(\ j G j) = 0. Indeed, for almost x, w converges to w(x) whenever r( ) 0 and x. Take such an x and assume that x / j G j. Then, for every j there exsts x j k such that x (xj k, δj R) and (δ j R) p w (x j k,δj R). Ths s a contradcton snce (δj R) p w (x j k,δj R) 0 when j. Note also that there exsts 0 < A < such that for all j, k and ball j k of type, Indeed, let j k be of type because x j k doublng, we get be of type. There exsts j l (δ j R) p w j > A. (6) k / G j. Hence j k (xj l and j l must, δ j ( + δ )R). Snce wdµ s such that x j k j l w( j l ) w ( (x j l, δ j ( + δ )R) ) C ( + δ ) s w ( (x j l, δ j R) ) C ( + δ ) s ( + d(xj l, x j k ) ) s w( j δ j k R ) C ( + δ ) s w( j k ) where s = log C and C s the doublng constant of wdµ. On the other hand, snce dµ s doublng µ( j l ) C ( + δ) s µ((x j, δ j ( + δ)r)) C ( + δ) s µ( j k ). l Snce j l s of type, we obtan (δ j R) p w j C C ( + δ ) s ( + δ) s δ p (δ j R) p w( j l ) k > C C ( + δ ) s ( + δ) s δ p. Thus we get (6) wth A = C C ( + δ ) s ( + δ) s δ p. From all these facts we deduce that ( u p + w u p )dµ ( u p + w u p )dµ N j, k: j j kof type k C mn((δ j R) p, w N j ) u p dµ j, k: j k k of type j k C N A u p dµ j, k: j k of type (δ j R) p j k

13 C ( ) p R N A mn R p u p dµ. j δ j R We used Fefferman-Phong nequalty n the second estmate, (6) n the penultmate one, and that the ( j k ) of type cover up to a µ null set n the last one. It remans p R to estmate mn j R j from below wth Rj = δ j R. Let α < be such that w A α the uckenhoupt class. Then for any ball and measurable subset E of we have Applyng ths to E = j k ( we w ) C and we obtan ( ) p R = Rp w R j R p j w j k ( ) α µ(e). µ() w j k w w R p j k w w ( CR p µ( j k w ) µ() ) α ( ) s(α ) CR p Rj w R where we used Lemma.3. Ths yelds mn j ( R R j ) p C(R p w ) β wth β = The lemma s proved. 4. Calderón-Zygmund decomposton We now proceed to establsh the followng Calderón-Zygmund decomposton: p p+s(α ). Proposton 4.. Let be a complete Remannan manfold satsfyng (D) and (P l ) for some l <. Let l p <, V A, f Ẇ and α > 0. Then, one can p,v fnd a collecton of balls ( ), functons g and b satsfyng the followng propertes: f = g + b (7) g + V g Cα p ( f p + V f p, (8) supp b and ( b l + V b l + R l b l )dµ Cα l µ( ), (9) µ( ) Cα p ( f p + V f p )dµ, (0) N, () 3

14 where N depends only on the doublng constant, and C on the doublng constant, p, l and the A constant of V. Here, R denotes the radus of and gradents are taken n the dstrbutonal sense on. Remark 4.3. It follows from the proof of Proposton 4. that the functon g s Lpschtz wth the Lpschtz constant controlled by Cα (see page 6 below). Proof of Proposton 4.. Let Ω be the open set {x ; ( f l + V f l )(x) > α l }. If Ω s empty, then set g = f and b = 0. Otherwse, the maxmal theorem Theorem.4 yelds µ(ω) Cα p ( f p + V f p )dµ <. () In partcular Ω as µ() =. Let F be the complement of Ω. Snce Ω s an open set dstnct of, let ( ) be a Whtney decomposton of Ω ([9]). That s, the balls are parwse dsjont and there s two constants C > C >, dependng only on the metrc, such that. Ω = wth = C are contaned n Ω and the balls ( ) have the bounded overlap property;. r = r( ) = d(x, F ) and x s the center of ; 3. each ball = C ntersects F (C = 4C works). For x Ω, denote I x = { : x }. y the bounded overlap property of the balls, we have that I x N. Fxng j I x and usng the propertes of the s, we easly see that r 3 r j 3r for all I x. In partcular, 7 j for all I x. Condton () s nothng but the bounded overlap property of the s and (0) follows from () and (). We remark that snce V A, Proposton. and Proposton., pont 3 yeld V l A. Applyng Lemma 4., we obtan ( f l + V f l )dµ C mn((v l ), R l ) f l dµ. (3) We declare of type f (V l ) R l and of type f (V l ) < R l. Let us now defne the functons b. Let (χ ) be a partton of unty on Ω assocated to the coverng ( ) so that for each, χ s a C functon supported n wth χ + R χ C. Set b = { fχ, f s of type, (f f )χ, f s of type. If s of type, then t s a drect consequence of the Poncaré nequalty (P l ) that ( b l + R l b l )dµ C f l dµ. 4

15 As f l dµ α l µ( ) we get the desred nequalty n (9). For V b we have V b l dµ = V (f f )χ l dµ ( ) C V f l dµ + V f l dµ ( ) C ( V f l ) µ( ) + C(V l ) ( f l ) µ( ) ( ( ) ) C α l µ( ) + f l + V f l µ( ) Cα l µ( ). We used that F, Jensen s nequalty and (3), notng that s of type. If s of type, R l b l dµ R l f l C ( f l + V f l )dµ. As the same ntegral but on s controlled by α l µ( ) we get R l b l dµ Cα l µ( ). Snce b = χ f +f χ we obtan the same bound for b l dµ. Notng that F and s of type, we easly deduce that V b l Cα l µ( ). Set g = f b where the sum s over both types of balls and s locally fnte by (). It s clear that g = f on F = \ Ω and g = f χ on Ω, where j means that we are summng over cubes of type j. Let us prove (7). Frst, by the dfferentaton theorem, V f α almost everywhere on F. Next, as we explaned before, V A mples V l RH and therefore V C((V l ) ) l. l Therefore Ω V g dµ V f C ((V l ) ) f l ) l µ( ). Now, by constructon of the type balls and the L l verson of Fefferman-Phong nequalty, (V l ) f l C( f l + V f l ) Cα l. It comes that V g dµ C α l µ( ) C α l Ω ( f l + V f l )dµ. Combnng the estmates on F and Ω, we obtan the desred bound for V g dµ. We fnsh the proof by estmatng g and g l. Observe that g s a locally ntegrable functon on. Indeed, let ϕ L wth compact support. Snce d(x, F ) R for x supp b, we obtan ( b ϕ dµ b ) dµ R 5 ( ) sup x d(x, F ) ϕ(x).

16 If s of type b dµ µ( R l Cµ( l Cαµ( ). b l dµ R l f l dµ We used the Hölder nequalty, (P l ) and that F, q beng the conjugate of q. If s of type, b dµ µ( b l l dµ Cαµ( R R l ). ( ) Hence b ϕ dµ Cαµ(Ω l sup d(x, F ) ϕ(x). Snce f L loc, we conclude x that g L loc. Thus g = f b. It follows from the L l estmates on b and the bounded overlap property that b C ( f l + V f l ). l As g = f b, the same estmate holds for g l. Next, a computaton of the sum b leads us to g = F ( f) f χ (f f ) χ. Set h j = j (f f ) χ and h = h + h. Then g = ( f) F f χ (h h ) = ( f) F + f χ h. y defnton of F and the dfferentaton theorem, g s bounded by α almost everywhere on F. y already seen arguments for type balls, f CαR. Therefore, f χ C α CNα. It remans to control h. For ths, note frst that h vanshes on F and the sum defnng h s locally fnte on Ω. Then fx x Ω. Observe that χ (x) = 0 and by defnton of I x, the sum reduces I x. For all I x, we have f(x) f Cr α. Hence, we have for all j I x, (f(x) f ) χ (x) = (f(x) f ) χ (x) = (f j f ) χ (x). I x I x We clam that f j f Cr j α wth C ndependent of, j I x and x Ω. Indeed, we use that and j are contaned n 7 j, Poncaré nequalty (P l ), the comparablty of r and r j, and that F. Snce I x has cardnal bounded by N, we are done. We conclude that h Cα and nterpolatng g l and g, we therefore fnsh the proof. 6

17 Proposton 4.4. Let be a complete Remannan manfold satsfyng (D). Let V A. oreover assume that admts a Poncaré nequalty (P p ) for some < p <. Then, Lp() Ẇ Ẇ s dense n Ẇ.,V p,v p,v Proof. Theorem.8 proves that admts a Poncaré nequalty (P l ) for some l < p. Let f Ẇ. For every n N, consder the Calderón-Zygmund decomposton p,v of Proposton 4. wth α = n. Take a compact K of. We have f g n l dµ = b K K ( S l dµ ) = b S l dµ K C f f K l d(x, F R l n ) l dµ + C f K l d(x, F R l n ) l dµ C sup x K(d(x, F n )) l C sup x K(d(x, F )) l ( f l + V f l )dµ n l µ( ) Cn l p ( f p p + V f p p ). Lettng n, we get that f g K n l dµ 0. Hence (f g n ) converges to 0 when n n the dstrbutonal sense. Let us check that (V (f g n )) n s bounded n L p. Indeed, V (f gn ) p dµ V f p dµ + V p f p dµ Ω n Ω n V f p dµ + ) p ((V l ) f l l µ( ) Ω n V f p dµ + Cn p µ(ω n ) Ω n C( f p p + V f p p ). Smlarly, f g n p dµ = f g n p dµ C Ω n f p dµ + Cn p µ(ω n ) C. Ω n Thus, ( f g n ) n s bounded n L p. So (f g n ) n s bounded n Ẇ. Snce p,v s reflexve Proposton.6, there exsts a subsequence, whch we denote Ẇ p,v also by (f g n ) n, convergng weakly n Ẇ to a functon h. The unqueness of p,v the lmt n the dstrbutonal sense yelds h = 0. y azur s Lemma, we fnd a Lp() s the set of all Lpschtz functons on. 7

18 sequence (h n ) of convex combnatons of (f g n ) such that h n = n k= a n,k(f g k ), a n,k 0, n k= a n,k =, that converges to 0 n Ẇ. Snce h p,v n = f l n and V h n = V (f l n ) wth l n = n k= a n,kg k, we obtan l n f n Ẇ and the n p,v proposton follows on notng that g n, hence l n, also belongs to Lp() Ẇ.,V 4.3 Estmates for subharmonc functons Fx an open set Ω. A subharmonc functon on Ω s a functon v L loc (Ω) such that v 0 n D (Ω). Lemma 4.5. Let be a Remannan manfold satsfyng (D) and (P ). Let R > 0 and x 0 be a pont such that a neghborhood of (x 0, 4R) s contaned n. Suppose that f s a non-negatve subharmonc functon defned on ths neghborhood. Then, there s a constant C > 0 ndependent of f, x 0, R such that sup f(x) C x (x 0,R) ( (x 0,4R) f (y)dµ(y) It readly follows from Lemma.3 that for all r > 0, < η < 4, there s C > 0 such that ( sup f(x) C f r r (y)dµ(y). (5) x (x 0,R) (x 0,ηR) Proof. In [44], Theorem 7., ths lemma s stated for Remannan manfolds wth non-negatve Rcc curvature. The proof reles on the followng propertes of the manfold. Frst, the Harnack nequalty for non-negatve harmonc functons whch holds for complete Remannan manfolds satsfyng (D) and (P ) (see [9]). Secondly, the Poncaré nequalty (P ). Fnally, the Caccoppol nequalty for non-negatve subharmonc functons Lemma 7. n [44] whch s vald on any complete Remannan manfold. We then get ths lemma under the hypotheses (D) and (P ). Other forms of the mean value nequalty for subharmonc functons stll hold f the volume form s replaced by a weghted measure of uckenhoupt type. ore precsely, Lemma 4.6. Consder a complete Remannan manfold satsfyng (D) and (P ). Let V A and f a non-negatve subharmonc functon defned on a neghborhood of (x 0, 4R), 0 < s < and < η < 4. Then for some C dependng on the A constant of V, s (and ndependent of f and x 0, R), we have ( C sup f(x) x (x 0,R) V ((x 0, ηr)) (x 0,ηR) V f s s dµ. Here V (E) = E V dµ. As A weghts have the doublng property we have V (x0,ηr) V (x0,r) and the nequalty above s the same as (4) V (x0,r)( sup f s ) C(V f s ) (x0,ηr). (6) (x 0,R) 8

19 Proof. Snce V A, there s t < such that V A t. Hence for any non-negatve measurable functon g we have Applyng (5) wth r = s t yelds ( g (x0,ηr) C V ((x 0, ηr)) = C ( (V g t ) (x0,ηr) f(x) C ( ) (f s t t s )(x0,ηr) (x 0,ηR) t ( V (x0,ηr) V g t t dµ ) t. C ( ) (V f s ) ( ) s (x0,ηr) V s (x0,ηr). Corollary 4.7. Let be a complete Remannan manfold satsfyng (D) and (P ). Let V RH r for some < r, 0 < s < and < η 4. Then there s C 0 dependng only on the RH r constant of V, s such that for any ball (x 0, R) and any non-negatve subharmonc functon defned on a neghborhood of (x 0, 4R) we have Proof. We have ( ((V f s ) r ) (x0,r) r C(V f s ) (x0,ηr). ( ((V f s ) r ) (x0,r) r C ( ) (V r r ) (x0,r) sup (x 0,R) f s CV (x0,r) sup (x 0,R) f s C(V f s ) (x0,ηr). The second nequalty uses the RH r condton on V and the last nequalty s (6). 5 axmal nequaltes Ths secton s devoted to the proof of Theorem.. Let < q and V RH q. The followng lemma s classcal n an Eucldean settng [7], [37] (see also [3]). Lemma 5.. Let be a complete Remannan manfold. We assume that V L loc () s not dentcally 0. Let u C0 (). Then V u dµ ( + V )u dµ, u dµ ( + V )u dµ. 9

20 Proof. Let us prove the estmate for V u. Take p n : R R a sequence of C functons such that p n C, p n(t) 0 and p n (t) sgn(t) for every t R. Usng the Lebesgue convergence theorem we see that sgn(u) udµ = lm p n (u) udµ = lm u p n n n(u)dµ 0. If u + V u = f, we get V u dµ sgn(u)( + V )udµ = f sgn(u)dµ Ths gves the desred estmaton for V u. The estmate for u follows from that of V u snce u + V u = f. f dµ. Let D (H) = {u L loc ; V u L loc, ( + V )u L }. One can easly check that C0 s dense n D (H) ([4] for a proof n the Eucldean parabolc case) thanks to the Kato nequalty on manfolds ([], Theorem 5.6). Thus the above estmates for V u and u stll holds for any u D (H). Lemma 5. shows that D (H) = {u L loc ; u L, V u L } equpped wth the topology defned by the sem-norms for L loc, u and V u. We have therefore obtaned Theorem 5.. The operator H a pror defned on L 0 () the set of compactly supported bounded functons defned on extends to a bounded operator from L () nto D (H). Denotng agan H ths extenson, V H s a postvty-preservng contracton on L () and H s a contracton on L (). Proposton 5.3. Assume that satsfes (D) and (P ). Let f L (). Then there s unque of soluton of the equaton u + V u = f n the class L () D (H). In partcular, f u C 0 () and f = u + V u, then u = H f. Proof. Assume u + V u = 0, then for ɛ > 0 we have u + V u + ɛu = ɛu. As u L (), we can wrte u ( + ɛ) (ɛ u ) = ( ɛ + ) u. Usng the upper bound of the kernel of ( ɛ + ) whch follows from (D) and (P ), and takng lmts when ɛ 0 we get u = 0. Corollary 5.4. Assume (D) and (P ). Then equaton () holds. Proof. If u C 0 () and f = u + V u, then V u = V H f and u = H f by the proposton above. Applyng Theorem 5. we get V u u + V u and u u + V u. We now gve the followng crteron for L p boundedness: Theorem 5.5. ([7]) Let be a complete Remannan manfold satsfyng (D). Let r 0 < q 0. Suppose that T s a bounded sublnear operator on L r 0 (). Assume that there exst constants α > α >, C > 0 such that ( T f ) { q 0 q ( 0 C T f ) r 0 r 0 + (S f )(x) }, (7) α 0

21 for any ball, x and all f L 0 () wth support n \ α, where S s a postve operator. Let r 0 < p < q 0. If S s bounded on L p (), then, there s a constant C such that T f p C f p for all f L 0 (). Note that the space L 0 () can be replaced by C 0 (). Now we use the L estmate () and Theorem 5.5 to get Theorem 5.6. Let be a complete Remannan manfold satsfyng (D) and (P ). Consder V RH q, wth q >. Then, there exsts r > q, such that V H and H defned on L () by Theorem 5. extend to L p () bounded operators for all < p < r. Proof. y dfference, t suffces to prove the theorem for V H. We know that ths s a bounded operator on L (). Let r be gven by the self-mprovement of the reverse Hölder condton of V. Fx a ball and let f L () wth compact support contaned n \ 4. Then u = H f s well-defned n V and s a weak soluton of u + V u = 0 n 4. Snce u s subharmonc (cf secton 8.), we can apply Corollary 4.7 wth V, f = u and s =. Thus (7) holds wth T = V H, r 0 =, q 0 = r, S = 0, α = and α = 4. Hence, Theorem 5.5 asserts that T = V H s bounded on L p () for < p < r. Proof of Theorem. Let u C 0 () and f = u + V u. Proposton 5.3 shows that u = H f. Snce V RH q, Theorem 5.6 shows that V H and H have bounded extensons on L p () for < p < q + ɛ for some ɛ > 0 dependng on V. Ths means that V u p + u p f p whch s the desred result. 6 Complex nterpolaton We shall use complex nterpolaton to obtan pont of Theorem.3. Ths method s based on the boundedness of magnary powers of H and of the Laplace-eltram operator. Then we use Sten s nterpolaton theorem to prove the boundedness of H on < p < nf(p 0, (q + ɛ)) and V H on L p () for < p < (q + ɛ) and therefore obtan pont of Theorem.3. Let y R, the operator H y s defned va spectral theory. One has H y =. Theorem 6.. Let be a complete Remannan manfold satsfyng (D) and assume that the heat kernel verfes the followng upper bound: for all x and t > 0 p t (x, x) C µ((x, t)). (8) Let V be a non-negatve locally ntegrable functon on. Then for all γ R, H γ has a bounded extenson on L p (), < p <, and for fxed p ts operator norm does not exceed C(δ, p)e δ γ for some δ > 0.

22 Remark 6.. The operator norm s far from optmal but suffcent for us. Nevertheless, as ponted out by the referee, the operator norm can be mproved to C( + γ ) s/ where s = log C d. Ths can be checked by a careful proof readng of [53], Theorem. Also a stronger result can be found n [4]. Proof of Theorem 6.. For V = 0, ths follows from the unversal multpler theorem for arkoven sem groups (Corollary 4, p. n [56]). However, the followng proof works for all V. Indeed, the remark after Theorem 3. n [6] apples to H: H has a bounded holomorphc functonal calculus on L () n any sector argz < θ, 0 < θ < π and the kernel h t (x, y) of e th has a Gaussan upper bound. Ths follows from the domnaton of e th by e t, (D) and (8). We have h t (x, y) C µ((x, d (x,y) t)) e c t for every t > 0, x, y. Thus a varant of Theorem 3. n [6] (see page 04 there) shows that H has a bounded holomorphc functonal calculus on L p () n any sector argz < µ, π < µ π for < p <. Ths mples H γ p p C(p, µ) sup argz <µ z γ C p,µ e γ µ. Lemma 6.3. The space D = R(H) L () L () s dense n L p () for < p <. Proof. Same proof as that of Lemma 6. n [3]. Proposton 6.4. Assume that satsfes (D) and (P ). Let V RH q for some q <. Then, for 0 < α <, there exsts ɛ > 0 such that the operators α H α, V α H α are bounded on L p () for < p < α q + ɛ. Proof. From Theorem 6., we have that γ and H γ are L p () bounded for < p < and γ R. oreover, Theorem. asserts that H and V H are L p () bounded for < p < q+ɛ for some ɛ > 0. It follows from Sten s nterpolaton theorem [54] that α H α, V α H α are bounded on L p () for < p < (q + ɛ) (see [3] for α detals). We can now prove pont of Theorem.3. Fx < p < (q + ɛ). Let u C 0 (). Snce u V, f = H u s well-defned. We assume that f L p (), otherwse there s nothng to prove. Applyng Proposton 6.4 to V, t comes that V u p C p f p. The L p () boundedness of the Resz transform whch holds for all < p < p 0 wth p 0 > on a complete Remannan manfold satsfyng (D) and (P ) and agan Proposton 6.4 yeld u p C(p) H f p C (p) f p for < p < nf(p 0, (q + ɛ)) and fnshes the proof. Remark 6.5. Ths nterpolaton argument also gves us a proof of the L p () boundedness of H and V H for < p < for all non zero V L loc ().

23 7 Proof of Theorem.4 The proof s smlar to that of pont of Theorem. n [3] wth some modfcatons. We wrte t for the sake of completeness. Assume that < l <. Let f Lp() Ẇ. y the spectral theory we have,v Ẇ l,v H f = c He th f dt 0 where c = π/. It suffces to obtan the result for the truncated ntegrals R... wth ɛ bounds ndependent of ɛ, R, and then to let ɛ 0 and R. For the truncated ntegrals, all the calculatons are justfed. We thus consder that H s one of the truncated ntegrals but we stll wrte the lmts as 0 and + to smplfy the exposton. As f does not belong to C0 (), we have to gve a meanng to He th f for t > 0. Take η r a smooth functon on, 0 η r, η r = on a ball of radus r > 0, η r = 0 outsde and η r C. For ϕ r C 0 (), f He th ϕdµ = = lm η r fhe r th ϕdµ = η r f. e th ϕdµ + f η r. e th ϕdµ + η r f V e th ϕdµ = I r + II r + III r. We used Fubn and Stokes theorems. Note that xh t (x, y) e γ d (x,y) t dµ(x) C tµ((y,. Ths s due to the Gaussan upper estmate of the kernel h t)) t of e th and that of t h t under (D) and (P ) (see [0], Lemma.3, for the heat kernel p t of e t ). Snce f L () then I r f. e th ϕdµ. Snce f s Lpschtz, II r 0. We also have V (x)h t (x, y) e γ d (x,y) C t dµ(x) µ((y, and V f L (). Thus t)) III r fv e th ϕdµ. Ths proves that He th f s defned as a dstrbuton by He th f, ϕ = f. e th ϕdµ + V fv e t H ϕdµ. Therefore, ntegratng n t yelds H f, ϕ = f, H ϕ + V f, V H ϕ. We return to the proof of Theorem.4. Apply the Calderón-Zygmund decomposton of Lemma 4. to f at heght α and wrte f = g + b. For g, we have ({ µ x ; H α }) g(x) > 9 H 3 α g dµ 9 ( g + V g )dµ α 3

24 C ( f l + V α l f l )dµ. We used a smlar argument as above to compute H g (see [4]) and the L estmate follows. For the last nequalty we used (8) of the Calderón-Zygmund decomposton and that l <. The argument to estmate H b wll use the Gaussan upper bound of h t. As we mentoned above, under our assumptons we have the Gaussan upper bound for the kernel of e th and by analytcty for He th. As b s supported n a ball and ntegrable He th b s defned by the convergent ntegral t th t (x, y)b (y)dµ(y). Let r = k f k R < k+ (R s the radus of ) and set T = r 0 He th dt and U = r He th dt. It s enough to estmate ({ }) A = µ x ; T b (x) > α 3 and Frst, ({ }) = µ x ; U b (x) > α. 3 A µ( ({ ) + µ x \ }) ; T b (x) > α, 3 and by (0), µ( ) C α ( f l + V l f l )dµ. For the other term, we have ({ µ x \ }) ; T b (x) > α C h 3 α wth h = ( ) c T b. To estmate the L norm, we dualze aganst u L () wth u = : u h = A j where A j = C j ( ) j= T b u dµ, C j ( ) = j+ \ j. y nkowsk ntegral nequalty, for some approprate postve constants C, c, T b L (C j ( )) r 0 He th b L (C j ( )) dt. y the well-known Gaussan upper bounds for the kernels of the th, t > 0, vald snce we have (D) and (P ) He th C cd b (x) (x,y) t µ((y, t)) e t b (y) dµ(y). 4

25 Now y supp b, that s, and x C j ( ), hence one may replace d(x, y) by j r n the Gaussan term snce r R. Also f x denotes the center of, we have µ((x, t)) µ((y, t)) = µ((x, t)) µ((x, r )) y (D) and Lemma.3 as t r, we have µ((x, t)) µ((y, t)) C(r t )s. µ((x, r )) µ((y, r )) µ((y, r )) µ((y, t)). Usng the estmate (9), b cαr µ( ), and µ( ) µ((x, r )), t comes that He th b (x) C t µ((x, t)) ( r t Cr t ( r ) s e c4j r t t b dµ ) s e c4j r t α. Thus He th b L (C j ( )) Cr ( r ) s e c4j r t t t µ( j+ α. Pluggng ths estmate nsde the ntegral, we get T b L (C j ( )) Cαe c4j µ( j+. Now remark that for any y and any j, ( ( u u ( ( s(j+) µ( ) ( u )(y). C j ( ) j+ Applyng Hölder nequalty, one obtans Averagng over yelds A j Cα sj e c4j µ( ) ( ( u )(y). A j Cα sj e c4j ( ( u )(y) dµ(y). Summng over j and, t follows that u h dµ Cα (y) ( ( u )(y) dµ(y). Usng fnte overlap () of the balls and Kolmogorov s nequalty, one obtans u h dµ C Nαµ( u. 5

26 Hence ({ µ x \ }) ; T b (x) > α Cµ( 3 ) C α l ( f l + V f l )dµ by () and (0). It remans to handle the term. Usng functonal calculus for H one can compute U as r ψ(r H) wth ψ the holomorphc functon on the sector arg z < π gven by ψ(z) = e tz z dt. It s easy to show that ψ(z) C z e c z, unformly on subsectors arg z µ < π. The (P l ) Poncaré nequalty gves us f s of type b l l CR l f l dµ CRα l l µ( ). If s of type b = (b (b ) ) + (b ). (9) Therefore usng the type property of and also (9) yeld ( ) b l dµ l b (b ) l + µ( ) b dµ l CRµ( l ) l b l dµ + Cµ( )R l ( f l + V f l )dµ CRµ( l ) l f l dµ + Cµ( )R l ( f l + V f l )dµ Cα l R l µ( ). Hence b l l Cαl Rµ( l ). We nvoke the estmate ( ψ(4 k H)β k β k. (30) k Z l k Z l Indeed, by dualty, ths s equvalent to the Lttlewood-Paley nequalty ( ψ(4 k H)β β l. k Z Ths s a consequence of the Gaussan estmates for the kernels of e th, t > 0 (ths was frst proved n [5] usng the vector-valued verson of the work n [5]. See [] or [6] for a more general argument n ths sprt or [39] for an abstract proof relyng on functonal calculus). To apply (30), observe that the defntons of r and U yeld U b = ψ(4 k H)β k k Z 6 l

27 wth β k = r.,r = k b Usng the bounded overlap property (), one has that Usng R r, ( l β k C ( k Z ( l b l )dµ Cα l r l b l )dµ. r l µ( ). Hence, by (0) ({ }) µ x ; U b (x) > α C 3 µ( ) C α l ( f l + V f l )dµ. Thus, we have obtaned ) µ ({x ; H f(x) > α} C ( f l + V α l f l )dµ for all f Lp() Ẇ Ẇ. l,v l,v oreover, usng the densty argument of Proposton 4.4 we extend H bounded operator actng from Ẇ to L l,. We already have l,v to a H f f + V f. Snce V A mples V RH Proposton., we see from Corollary.7 that ( ) H f p C p f p + V f p (3) for all l < p and f Ẇ p,v. If l =, we take < p <. There exsts ɛ > 0 such that < + ɛ < p. The same argument works replacng l = by + ɛ. 8 Proof of pont. of Theorem.3 We frst gve some estmates for the weak solutons of u + V u = 0. proceed to a reducton and then gve the proof of pont. of Theorem.3. Then we 7

28 8. Estmates for weak solutons Let be a complete Remannan manfold satsfyng (D) and (P ). Let = (x 0, R) denotes a ball of radus R > 0 and u a weak soluton of u + V u = 0 n a neghborhood of (x 0, 4R). y a weak soluton of u + V u = 0 n an open set Ω, we mean u L loc (Ω) wth V u, u L loc (Ω) and the equaton holds n the dstrbuton sense on Ω. Remark that under the Poncaré nequalty (P ) f u s a weak soluton, then u L loc (Ω). It should be observed that f u s a weak soluton n Ω of u + V u = 0 then u = V u + u (3) snce u = u, u + u (see [0]). In partcular, u s a non-negatve subharmonc functon n Ω. Hence the lemmas n subsecton 3 of secton 4 apply to u. In partcular sup u C(r) ( ) ( u r r ) (x0,µr) (33) (x 0,R) holds for any 0 < r < and < λ 4. We have also shown a mean value nequalty aganst arbtrary A weghts. We state some further estmates that are nterestng n ther own rght assumng V A. y splttng real and magnary parts, we may suppose u real-valued. All constants are ndependent of and u but they may depend on the constants n the A condton or the RH q condton of V when assumed, on the doublng constant C d and the Poncaré nequalty (P ). Let s be any real number such that µ() C( r µ( 0 ) r 0 ) s whenever = (x, r), x 0, r r 0 (s = log C d works). The proofs of the next 3 lemmas are as n [3], we skp them. Lemma 8.. For all λ < λ 4 and k > 0, there s a constant C such that and ( u ) λ ( u + V u ) λ C ( + R V ) k ( u ) λ. C ( + R V ) k ( u + V u ) λ. Lemma 8.. For all λ 4, k > 0, there s a constant C such that (RV ) ( u ) C ( + R V ) k (V u ) λ. Lemma 8.3. For all < λ 4, k > 0 and max(s, ) < r <, there s a constant C such that (RV ) ( u C ) ( + R V ) k ( u r ) r λ. The man tools to prove these lemmas are the mproved Fefferman-Phong nequalty of Lemma 4., the Caccoppol type nequalty whch holds on complete Remannan manfolds, Poncaré nequalty, subharmoncty of u, Lemma 4.6 and the orrey embeddng theorem wth exponent α = s r s = Log C, wth C the doublng constant ([34], Theorem 5., p. 3) to prove Lemma

29 For the remanng lemmas, we moreover assume that s of polynomal type: every ball of radus r > 0 satsfes and µ() cr σ, (L σ ) µ() Cr σ (U σ ) wth σ = d f r and σ = D for r and d D. Note that f (L σ ) holds then σ n where n s the topologcal dmenson of (see [50]). Recall that under (L σ ) and (U σ ), s = D works and that µ((x, r)) cr λ for all r > 0 wth any λ [d, D]. We also recall that the exponent p 0 s that appearng n Proposton.. Lemma 8.4. Assume V RH q. Let be a ball of radus R > 0. Set q = qσ f qσ < p 0 ( = ) and q arbtrary n ], p qσ q σ 0[ f not. Then for all k > 0 there s a constant C = C(σ) ndependent of such that ( ( u q ) q C ( + R V ) k ( ( u + V u ) 4. Lemma 8.5. Assume V RH q wth D q < p 0. Let be a ball of radus R > 0. Set q = qσ f qσ < p 0 and q arbtrary n ]q, p 0 [ f not. Then, there s a constant C = C(σ) such that ( ) ( u q q ) C ( ) ( u ) 4, We gve the proofs of Lemma 8.4 and 8.5 snce they are not exactly the same as the one n the Eucldean case. efore the proof of Lemma 8.4, we need the followng theorem for the boundedness of the Resz potental. Theorem 8.6. ([7]) Let be a complete Remannan manfold satsfyng (D) and (P ). oreover, assume that satsfes µ() cr λ (L λ ) for every x and r > 0. Then ( ) s L p L p bounded wth < p, p < and p = λp, that s, λ p ( ) f p C(p, λ) f p. Proof. In [7], Chen proves ths theorem for Remannan manfolds wth non-negatve Rcc curvature. Hs proof stll works under our hypotheses. The propertes that he used for these manfolds are frst the lower and upper gaussan estmates for the heat kernel whch holds on Remmanan manfolds satsfyng (D) and (P ). Secondly, he appled an argument from the proof of the L p L p boundedness of the Resz potental n the Eucldean case ([55], Chapter V, Theorem ) whch remans true snce we have (D), (P ) and (L λ ) wth λ n = dm. 9

30 Proof of Lemma 8.4. Frst note that f q σ σ+ for us) follows by a mere Hölder nequalty. Henceforth, we assume q > then q and the concluson (useless σ. Also, σ+ by Lemma 8., t suffces to obtan the estmate wth k = 0. Let us assume µ = 4 for smplcty of the argument. Let v be the harmonc functon on 4 wth v = u on (4) and set w = u v on 4. Snce w = 0 on (4), the fact that an harmonc functon mnmses Drchlet ntegral among functons wth the same boundary mples ( 4 w ) ( 4 u. y the ellptc estmate for the harmonc functon v ([4], Theorem.), we have for p < p 0 ( v p) p C( v ) C( u. (34) 4 4 Let < ν < λ < 4 and η be a smooth non-negatve functon, bounded by, equal to on ν wth support contaned n λ and whose gradent s bounded by C. As R w = u = V u on 4, we have It comes that (wη) = V uη + w η + dv(w η) on. (wη)(x) = ( ) ( )(wη)(x) Let us begn wth = ( ) ( ) ( V uη)(x) + ( ) ( ) ( w. η)(x) + ( ) ( dv(w η))(x) = I + I + I 3. I 3 = ( ) ( ) dv( w η)(x) = ( ( ) )( ( ) ) ( w η)(x). Let η be a smooth functon, bounded by, equal to on λ wth support contaned n λ wth λ < 4 and whose gradent s bounded by C. The Resz transform R ( ) s L p () bounded for < p < p 0. y dualty, ( ( ) ) s L p () bounded for p 0 < p <. Hence for < p < p 0 ( ( I 3 p p dµ C wη p η p p dµ C R ( (wη p ) p dµ. We used the Sobolev nequalty whch holds under (D), (P ) and µ((x, r)) cr σ for all r > 0 wth p σ < p defned by p σ = σp that s (p σ+p ) = p (see [50]). We use the L q L q σ boundedness of the Resz potental ( ) and the L p boundedness of the Resz transform ( ) for < p < p 0 to get the estmates for I and I. Frst for I, we have for all p < p 0 ( ( I p p dµ C ( ) ( (wη ). η) p dµ 30 p

31 ( C R C ( R = C ( R (wη p σ ) p σ dµ (wη p σ ) p σ dµ (wη p σ ) p σ dµ. Now, t remans to look at I. Take p = q σ f q σ < p 0 and f not any < p < p 0. It follows that ( I p p dµ C ( p σ V uη p σ dµ Cµ() p σ ( λ V q q dµ sup u µ snce p σ q n the two cases. Usng the RH q condton on V, we obtan ( I p dµ ) p Cµ( p σ V dµ sup u. (35) µ Now, f λ < γ < 4, the subharmoncty of u and Lemma 4.6 yeld V dµ sup u C V dµ ( u dµ. λ λ It follows from Lemma 8. and (U σ ) that ( Ip dµ p Therefore, we showed that ( (wη) p p dµ γ λ γ ( C (wη p ) p dµ + Cµ() p R ( Cµ( p V 4 u dµ. ( 4 V u dµ. We repeat the same process and after a fnte teraton (K = (σ[ p ] + ) tmes), usng (U σ) we get ( ( w q q dµ C 4 ( w dµ) + C 4 V u dµ. We derve therefore the desred nequalty for u from the estmates obtaned for v and w. Proof of Lemma 8.5. Snce V RH q and q D, we may assume q > D by selfmprovement. Let σ = d f R and σ = D f R. We apply the same arguments as n the proof of the prevous lemma. The only dfference s that snce q > s = D, we use Lemma 8.3 wth k = 0, r = q, and s = D nstead of Lemma 8. n the estmate for the term I. We then obtan ( q u q) C ( u q q (36) 4 3

32 where p = qσ f qσ < p 0 and f not we take any < p < p 0. Snce q < p 0, f we take p = q ]q, p 0 [ n (36) we can apply Lemma.3 and mprove the exponent q to. Thus, we get ( q u q) C ( u 4 Remark that when q > D, qσ = and therefore we have our lemma for any q < p < p A reducton It s suffcent to prove the L p boundedness of H and of V H for the approprate range of p. As we have seen n the ntroducton, the case < p does not need any assumpton on V. We henceforth assume p > and V A. y dualty, we know that H dv and H V are bounded on L p for < p <. Thus, f H s also bounded on L p, t follows that H dv and H V are bounded on L p. Recprocally, f H dv and H V are bounded on L p, then ther adjonts are bounded on L p. Thus, f F C0 (, T ), H dvf p = H H dvf p C( H dvf p + V H dvf p ) C F p where the frst nequalty follows from Theorem.4. y dualty, we have that H s bounded on L p. The same treatment can be done on V H. We have obtaned Lemma 8.7. Let be a complete Remannan manfold. If V A and p >, the L p boundedness of H s equvalent to that of H dv and H V, and the L p boundedness of V H s equvalent to that of V H V and V H dv. Hence, to prove Theorem.3, t suffces the L p H dv, H V, V H V, V H dv. boundedness of the operators: 8.3 Proof of pont. of Theorem.3 Proposton 8.8. Let be a complete Remannan manfold satsfyng (D) and (P ). Assume that V RH q for some q >. Then for < p < (q + ɛ), for some ɛ > 0 dependng only on V, f C 0 (, C) and F C 0 (, T ), V H V f p C p f p, V H dvf p C p F p. Proposton 8.9. Let be a complete Remannan manfold of polynomal type satsfyng (P ). Let V RH q for some q >. If q D < p 0, let p = q D. If q D p 0, we take any < p < p 0. Then for all f C 0 (, C) and F C 0 (, T ), H V f p C p f p, H dvf p C p F p. 3