OFF-DIAGONAL UPPER ESTIMATES FOR THE HEAT KERNEL OF THE DIRICHLET FORMS ON METRIC SPACES

Transcription

1 OFF-DIAGONAL UPPER ESTIMATES FOR THE HEAT KERNEL OF THE DIRIHLET FORMS ON METRI SPAES ALEXANDER GRIGOR YAN AND JIAXIN HU Absrac We give equivalen characerizaions for off-diagonal upper bounds of he hea kernel of a regular Dirichle form on he meric measure space, in wo seings: for he upper bounds wih he polynomial ail ypical for jump processes) and for he upper bounds wih he exponenial ail for diffusions) Our proofs are purely analyic and do no use he associaed Hun process onens Preliminaries and he main resuls General seup 2 Upper bounds of he hea kernel 2 2 Upper bound wih a polynomial ail 7 3 Upper bound wih an exponenial ail 2 3 The ail of he hea semigroup 2 32 Poinwise esimaes of he hea kernel 2 4 Appendix: Markovian properies 24 4 Maximum principle for weak soluions omparison lemmas for he resolven omparisons lemmas for he hea semigroup 3 References 34 Preliminaries and he main resuls General seup Le M, d) be a locally compac, separable meric space, and le µ be a Radon measure on M wih full suppor Le E, F) be a Dirichle form on L 2 M) := L 2 M, µ) Tha is, E is a closed, symmeric, non-negaive definie, bilinear form on a dense subspace F of L 2 M), which saisfies he Markov propery The closedness of he form E means ha F is a Hilber space wih respec o he E -inner produc, where E f, g) = E f, g) + f, g) Here, ) sands for he inner produc on L 2 M) The Markov propery means ha if f F hen he funcion f = maxmin f, ), ) is also in F and E f) E f) here and in he sequel we use he abbreviaion Ef) := Ef, f)) Le be he generaor of E, F), ha is, is a non-posiive definie self-adjoin operaor in L 2 M) wih domain dom ) F, and f, g) = Ef, g) for all f dom ), g F Dae: December 27 Revised March 28 AG was suppored by SFB 7 of he German Research ouncil DFG) JH was suppored by a Major Basic Research Gran of NSF Gran No 634) and by a visiing gran of SFB 7

2 2 GRIGOR YAN AND HU The generaor gives rise o he hea semigroup {P }, which is he family of bounded selfadjoin operaors in L 2 M), defined by ) P = e Obviously, he hea semigroup is conracive in L 2 M), ha is, P f f for all f L 2 M), where is he L 2 -norm, and srongly coninuous, ha is, P f f as for all f L 2 M) In addiion, he semigroup {P } is Markovian, ha is, if f ae, hen, for all >, 2) P f ae see [, Theorem 4, p 23]) The Markovian propery 2) allows o exend P f o all f L M) so ha P can be considered also as a conracion operaor from L M) o L M) onversely, given a srongly coninuous Markovian semigroup {P }, one recovers he corresponding Dirichle form by leing 3) E f) = lim f P f ), f, and by leing F be he se of hose f L 2 M) for which E f) < Le M) be he space of all coninuous funcions on M wih compac suppor The Dirichle form E, F) is called regular if F M) is dense boh in F in he E -norm) and in M) in he sup-norm The Dirichle form E, F) is called local if Ef, g) = for any f, g F wih disjoin compac suppors cf [, p6]) The hea semigroup or he Dirichle form) is called conservaive if P = ae for any > 2 Upper bounds of he hea kernel Definiion A family {p x, y)} > of measurable funcions on M M is called he hea kernel of E, F) if, for all f L 2 M), >, and µ-almos all x M, 4) P fx) = p x, y)fy)dµy) M The hea kernel does no have o exis bu if i exiss hen i is unique up o a se of measure zero) and saisfies he following properies, which follow immediaely from he corresponding properies of he hea semigroup: i) For all > and almos all x, y M, p x, y) and 5) p x, y) dµy) ii) For all, s > and almos all x, y M, 6) p +s x, y) = p x, z)p s z, y) dµz) iii) For all > and almos all x, y M, 7) p x, y) = p y, x) M M If he hea semigroup is conservaive hen 3) leads o he following relaion beween he hea kernel and he Dirichle form: for all f, g F, E f, g) = lim E f, g), By definiion, he suppor supp f of a funcion f L 2 M) is he se M \ Ω where Ω is he maximal open subse of M such ha f = ae in Ω

3 where 8) E f, g) = 2 M M UPPER ESTIMATES 3 f x) f y)) g x) g y)) p x, y) dµ x) dµ y) see [2]) Assuming ha he hea kernel exiss, we are ineresed in he following upper esimae: { )} UE) p x, y) min V ρ)), dx, y) V dx, y)) h, ρ) which is assumed o be rue for all > and almos all x, y M, wih some consan > Here and hroughou he paper, assume ha ρ : [, ] [, ] is a sricly increasing coninuous funcion such ha ρ) = and ρ ) = ; V : [, ) [, ) is an increasing funcion such ha V ) = and V r) > if r > ; h : [, ] [, ] is a sricly decreasing coninuous funcion such ha h) > and h ) = Seing r := d x, y), one can equivalenly sae UE) as follows:, if r ρ ), V ρ)) 9) p x, y) ) r V r) h, if r > ρ ) ρ) Indeed, he implicaion U E) 9) is obvious Now assume ha 9) holds and we will deduce UE) If r ρ ) hen, using he monooniciy of V and h, we obain V ρ)) V r) h ) ) r V r) h ρ) so ha he firs line in 9) implies UE) If r > ρ ) hen similarly ) r V r) h h ) ρ) V ρ )) so ha he second line in 9) implies UE) I is obvious ha UE) implies he on-diagonal upper esimae of p x, y): DUE) p x, y) V ρ)), for all > and almos all x, y M Le us sae wo well-known paricular cases of he esimae UE) In he boh cases, we assume ha ) V r) = r α and ρ ) = /β, for some posiive exponens α, β Example 2 Non-local Dirichle form) Le ) h s) = s β We claim ha UE) is equivalen o 2) p x, y) α/β + ) dx, y) α+β) /β for all > and almos all x, y M Indeed, UE) is equivalen o p x, y) V ρ )) + V r), hr/ρ))

4 4 GRIGOR YAN AND HU which for he seleced funcions V, ρ, h becomes 2 p x, y) α/β + r α r/ /β) β α/β + r α+β / α/β + r/ /β ) α+β The esimae 2) holds wih α = n and wih < β < 2 for he hea kernel of he operaor ) β/2 in R n where is he classical Laplace operaor), which a he same ime is he ransiion densiy of he symmeric β-sable process in R n The esimae 2) was shown in [7] o be rue in he following seing: he meric space M, d) is a subspace of some R n, he measure µ of a meric ball B x, r) see 5)) saisfies he esimae µ B x, r)) r α, for all x M and r >, and he Dirichle form is defined by E f, g) = f x) f y)) g x) g y)) J x, y) dµ x) dµ y) M M where J x, y) d x, y) α+β Here α, β are arbirary consans in he range α > and β, 2) Example 3 Local Dirichle form) Assume ha β > and se h s) = exp c s β/β )), for some c > We claim ha UE) is equivalen o ) ) dx, y) β/β ) 3) p x, y) α/β exp c Indeed, if r := d x, y) /β hen 3) is equivalen o /β p x, y) α/β, which is exacly he firs case of 9) Assume now r > /β Then 9) becomes r ) ) β/β ) 4) p x, y) r α exp c /β, which clearly implies 3) The converse implicaion 3) 4) follows from he inequaliy r ) ) β/β ) α/β ε r α exp ε /β, which is rue for any ε >, wih a big enough consan ε The purpose of his paper is o give some new equivalen characerizaions of U E) We emphasize ha he argumen in his paper is purely analyical, wihou recourse o he heory of Markov process Our main resuls show how one can obain he esimae UE) from he diagonal upper bound DU E) and some addiional condiions All known so far resuls have used some probabilisic condiions such as he firs exi ime from a ball, ec cf [2, 3, 6] For srongly recurren graphs, he reader may refer o [3] See [5] for an analyical approach on effecive-resisance meric spaces 2 The relaion f g means ha f g f for some posiive consan, for he specified range of he argumens of funcions f, g

5 UPPER ESTIMATES 5 As for he diagonal upper bound, here are pleny of various equivalen characerizaion of DU E) in erms of he Nash-ype inequaliy [6], [8], [8], he Faber-Krahn inequaliy [], he Sobolev inequaliy [9], he log-sobolev inequaliy [9], ec To explain he resuls, le us inroduce some noaion and erminology Le 5) Bx, r) := {y M : dy, x) < r} denoe a meric ball in M, d) onsider he following condiions: For all r and x M, V ) µ Bx, r)) V r) T ) T ) For all, r > and almos all x M, Bx,r) c p x, z) dµz) h For all, r, R >, y M and almos all x M, Bx,r) c By,R) ) r ρ) p x, z) dµz) V R) V r) h ) r ρ) In all condiions, is a posiive consan ha is independen on he variables in quesion and ha can ake differen values on difference occurrences The inegral in T ) should be undersood as follows: p x, z) dµz) := p x, z) Bx,r) c z) dµ z) Bx,r) c M Indeed, he funcion F x, z) = p x, z) Bx,r) c z) is obviously measurable joinly in x, z so ha by Fubini s heorem he inegral M F x, z) dµ z) is well-defined as a measurable funcion of x On he conrary, he inegral in T ) canno be undersood as he value on he diagonal {x = y} of he funcion G x, y) := p x, z) dµ z), By,r) c because a measurable funcion canno be resriced o a se of measure zero I is easy o show ha T ) can be equivalenly saed as follows: for all, r > and all x M, ) r 6) essup p x, z) dµz) h x Bx,r/2) Bx,r) c ρ) cf Remark 33 for he proof) The esimae 6) can be inerpreed as an upper bound for he funcion u, x) = P Bx,r) c, which is illusraed on Fig The esimae T ) can be reformulaed similarly and is illusraed on Fig 2 Our firs main resul Theorem 2, says ha, if he funcion ρ saisfies he doubling propery 3, funcions V and h are polynomial-like see 2)-25)), and he measure of he balls saisfies V ), hen UE) DUE) + T ) + T ) For example, he seing of he Example 2 maches he hypoheses of Theorem 2 A similar equivalence for U E) under he assumpions ) and )) was proved in [4] alhough insead of he condiions T ) and T ), wo alernaive hypoheses were used, which were saed in erms of he exi ime of he associaed jump process 3 A funcion f : [, ) [, ) is said o saisfy he doubling propery if here is a consan c > such ha 7) f2r) cfr) for all r

6 6 GRIGOR YAN AND HU u,x) h r ) Bx, /2r) Bx, r) x u,x)= u,x)= M Figure Funcion u, ) = P Bx,r) c VR) u,x) h Vr) r ) Bx, /2r) Bx, r) u,x)= By,R) M u,x)= Figure 2 Funcion u, ) = P Bx,r) c By,R) Our second main resul Theorem 3, reas he case when he Dirichle form is local In his case, one expecs he upper bound of sub-gaussian ype as in Example 3 onsider he following modificaion of he condiion U E): UE exp ) where p x, y) V ρ)) exp Φ c dx, y) { } s Φ s) = sup λ> ρ /λ) λ For example, under he condiions ) wih β >, we obain Φ s) = cs β β, and UE exp ) becomes 3) Le us inroduce he following weak version of he condiion T ): for any ε > here exiss K > such ha, for all r and such ha r Kρ ) and for almos all x M, T weak ) p x, z) dµz) ε Bx,r) c )), Equivalenly, T weak ) means ha T ) holds wih some unspecified) funcion h such ha h s) as s Theorem 3 says ha if E, F) is a regular, conservaive, local Dirichle form in

7 UPPER ESTIMATES 7 L 2 M), if all meric balls in M are precompac and saisfy V ), and if he funcions ρ and V are doubling, hen UE exp ) DUE) + T weak ) in his resul, we do no use he condiion T )) For he case ρ ) = /β, his equivalence was proved in [3], using he probabilisic approach The main ingrediens in he proof of Theorem 3 are Theorems 3, 34 ha are of independen ineres The main poin of hose heorems is ha he localiy of he Dirichle form allows o self-improve he condiion T weak ) hus leading o he following inequaliy Bx,r) c p x, z) dµz) exp Φ which is rue for all, r > and for almos all x M The proof of Theorems 3, 34 use he maximum principles for he resolven equaion Proposiion 46) and for he hea equaion Proposiion 4), hrough heir consequences orollary 45 and Lemma 48 To make he accoun self-conained, we presen in Appendix he analyic proofs of he maximum principles, which may be of heir own ineres Noaion Leers c,, K > and ε, ) denoe he consans whose values may change a each occurrence If A is a subse of M hen A c is is complemen, ha is, A c = M \ A If B = B x, r) is a ball in M, d) hen αb := B x, αr) Acknowledgemens The auhors are graeful o Takashi Kumagai for valuable discussions, which have led o improvemens of Theorem 34 c r )), 2 Upper bound wih a polynomial ail In his secion, we show ha UE) is equivalen o DUE)+T )+T ) under some addiional mild assumpions on V, h and ρ onsider he following condiions: ) V r 2 ) α r2 2), V r ) r ) V r 2 ) α2 r2 22) c, V r ) r ) hr 2 ) β r2 23), hr ) r ) hr 2 ) β2 r2 24) c, hr ) where each inequaliy is assumed o be rue for all < r < r 2 < and for some posiive consans c,, α, α 2, β, β 2 learly, if 2) and 22) hold simulaneously hen α 2 α, and if 23) and 24) hold simulaneously hen β β 2 Theorem 2 Le E, F) be a Dirichle form in L 2 M) wih he hea kernel p x, y) Assume ha he funcion ρ is doubling, and ha V and h saisfy 2)-22) and 23)-24), respecively, wih he addiional condiion ha 25) α β < [ α2 β 2 r ] + Assume also ha he volume of he meric balls in M saisfies V ) Then DUE) + T ) + T ) UE) Here [s] denoes by he ineger par of he number s The proof of Theorem 2 will be spli ino a series of Lemmas

8 8 GRIGOR YAN AND HU Lemma 22 Le E, F) be a Dirichle form wih he hea kernel p x, y) Assume ha he volume of he meric balls in M saisfies V ) Le funcion V be doubling and h saisfy 23)-24) Then UE) T ) + T ) + DUE) Proof The implicaion UE) DUE) is rivial Le us show ha UE) T ) Fix > and r > If ρ) r hen he monooniciy of h implies, for almos all x M, ) r p x, y) dµy) h, Bx,r) c ρ) where = h), which obviously maches T ) Assume now ha ρ) < r I follows from UE) and he monooniciy of V and h ha, for almos all x, z M such ha d z, x) s, 26) p x, z) ) s V s) h ρ ) Using 26), V ), he doubling propery of V, and 23)-24), we obain, for almos all x M, p x, z)dµ z) p x, z) dµz) Bx,r) c k= Bx,2 k+ r)\bx,2 k r) 2 k ) V 2 k r) h r ) µ Bx, 2 k+ r)\bx, 2 k r) ρ) k= V 2k+ r) 2 k ) V 2 k r) h r ρ) k= 2 k) ) β r h ρ) k= ) r h ρ) ondiion T ) is deduced from 26) and V ) as follows: p x, z) dµz) r Bx,r) c By,R) V r) h ρ) r V r) h ρ) This finishes he proof For any q, consider he following condiion H q ) p x, y) q V ρ)) hq ) µ By, R) Bx, r) c ) ) V R) ) dx, y), ρ) which should be rue for some q >, all >, and almos all x, y M learly DUE) is equivalen o H ), and 27) H ) + H q2 ) H q ) provided q < q 2 < Lemma 23 Assume ha T ) and T ) hold where ρ is doubling, V saisfies 2)-22), and h saisfies 24) If p x, y) saisfies H q ) where 28) q < α 2 β 2, hen i also saisfies H q+ )

9 UPPER ESTIMATES 9 Proof We need o prove ha, for any > and almos all x, y M, ) r 29) p 2 x, y) V ρ)) hq+, ρ) where r = 2 d x, y), which implies H q+) due o he doubling properies of funcions V, ρ and 24) If r ρ ) hen 29) follows from H q ) so ha we can assume r > ρ ) By he semigroup propery, we have, for all x, y M, p 2 x, y) = p x, z)p z, y) dµz) M 2) p x, z)p z, y) dµz) + p x, z)p z, y) dµz) Bx,r) c By,r) c On order o esimae he firs erm on he righ-hand side of 2) he second erm can be reaed similarly), spli i ino a good and a bad par: 2) p x, z)p z, y) dµz) = g, x, y) + b, x, y), Bx,r) c where 22) g, x, y) := and 23) b, x, y) := Le us firs esimae he good par z / By, r), 24) p z, y) Bx,r) c \By,r) By,r) p x, z)p z, y) dµz) p x, z)p z, y) dµz) I follows from H q ) ha, for almos all y M and ) r V ρ)) hq ρ) Subsiuing his ino 22) and using T ) we obain, for almos all x, y M, ) r g, x, y) V ρ)) hq p x, z) dµz) ρ) Bx,r) ) c r 25) V ρ)) hq+ ρ) In order o esimae he bad par, represen i in he form 26) b, x, y) = k= By,2 k r)\by,2 k+) r) p x, z)p z, y) dµz) see Fig 3) By condiion H q ) and 24), we see ha, for almos all y M and z By, 2 k r) \ By, 2 k+) r), ) 2 k+) r 27) p z, y) V ρ)) hq 2kβ 2 q ρ) V ρ)) hq ) r ρ)

10 GRIGOR YAN AND HU By,2 -k r) By,2 -k+) r) x y Bx,r) By,r) Figure 3 Esimaing he bad par Using 27), T ), and 22), we obain 28) By,2 k r)\by,2 k+) r) p x, z)p z, y) dµz) 2kβ 2 q ) r V ρ)) hq ρ) 2kβ 2 q ) r V 2 k r) V ρ)) hq h ρ) V r) 2kβ 2 q 2 kα ) 2 r V ρ)) hq+ ρ) p x, z) dµz) Bx,r) c By,2 k r) ) r ρ) here we have used he obvious fac ha he balls B x, r) and By, 2 k r) are disjoin) I follows from 26), 28), and 28) ha ) r ) 29) b, x, y) V ρ)) hq+ 2 kβ 2 q α2) r ρ) V ρ)) hq+ ρ) ombining 2), 25) and 29) we obain 29), which finishes he proof k= orollary 24 Under he hypoheses of Lemma 23, we have DUE) H q ) for any q < [ α2 β 2 ] + Proof Recall ha DUE) is equivalen o H ) Repeaedly applying Lemma 23, we obain he conclusion Lemma 25 Assume ha T ) holds where ρ is doubling, V saisfies 2)-22), and h saisfies 23)-24) Then DUE) and H q ) wih some q > α β imply ha ) dx, y) 22) p x, y) V d x, y)) h ρ) for all > and almos all x, y M Proof The argumen is similar o ha of Lemma 23 Le us prove ha, for all > and almos all x, y M, 22) p 2 x, y) ) r V r) h, ρ)

11 UPPER ESTIMATES where r = 2d x, y), which will imply 22) If r ρ) hen 22) immediaely follows from DUE) Assume in he sequel ha r > ρ ) As in he proof of Lemma 23, i suffices o show ha p x, z)p z, y)dµz) ) r Bx,r) c V r) h ρ) Wriing for simpliciy ρ insead of ρ ), we have p x, z)p z, y)dµz) p x, z)p z, y)dµz) Bx,r) c By,ρ) 222) + Bx,r) c By,2 k+ ρ)\by,2 k ρ)) p x, z)p z, y) dµz) see Fig 4) k= Bx,r) c By,2 k+ ρ) By,2 k ρ) Bx,r) By,ρ) y x I follows from p z, y) p x, z)p z, y)dµz) 223) By,ρ) Figure 4 Illusraion o he esimae 222) V ρ) ae and T ) ha V ρ) = V ρ) V ρ) = V r) h By,ρ) p x, z)dµz) Bx,r) c By,ρ) ) V ρ) r V r) h ρ ) r ρ p x, z)dµz) we have used here he fac ha he balls B x, r) and By, ρ) are disjoin) On he oher hand, H q ) and 23) imply ha, for almos all y M and z By, 2 k+ ρ) \ By, 2 k ρ), 224) p z, y) V ρ) hq Nex, by T ) and 2), 225) Bx,r) c By,2 k+ ρ) 2 k ) ρ ρ 2 kqβ V ρ) p x, z) dµz) V 2k+ ρ) h V r) 2 kα V ρ) V r) h ) r ρ ) r ρ

12 2 GRIGOR YAN AND HU Therefore, we obain from 224) and 225) ha, for almos all x, y M, Bx,r) c By,2 k+ ρ)\by,2 k ρ)) p x, z)p z, y) dµz) 226) k= k= k= V r) h 2 kqβ V ρ) 2 kqβ V ρ) r ρ ), Bx,r) c By,2 k+ ρ) 2 kα V ρ) V r) h r ρ p x, z) dµz) where he series converges due o qβ > α ombining 222), 223) and 226), we finish he proof Proof of Theorem 2 The implicaion )) UE) DUE) + T ) + T ) was obained in Lemma 22 Le us prove he converse, ha is, By orollary 24, we have [ ] α2 β 2 DUE) + T ) + T ) UE) DUE) + T ) + T ) H q ), [ ] β < α2 β 2 for any < q < + Since α +, we obain ha H q ) holds for some q > α β Hence, by Lemma 25, we obain 22) ombining 22) wih DUE) we obain UE) 3 Upper bound wih an exponenial ail The main purpose of his secion is o prove he upper bound for p x, y) wih he exponenial ail provided he Dirichle form E, F) is regular, conservaive, and local 3 The ail of he hea semigroup Here we do no assume he exisence of he hea kernel and work wih he funcion P B c, where B = B x, r) We use αb as a shorhand for B x, αr) as saed before The reader may refer o he definiion of P B in Secion 4 Theorem 3 Assume ha E, F) is a regular conservaive Dirichle form in L 2 M) and le all meric balls in M be precompac Le ρ : [, ) [, ) be any coninuous sricly increasing funcion wih ρ) =, ρ ) =, and le ρ saisfy he doubling propery Then he following condiions are equivalen i) For any ε, ) here exiss K > such ha, for any > and any ball B = Bx, r) wih r Kρ ), 3) P B c ε ae in 4 B ii) For any ε, ) here exiss K > such ha, for any > and any ball B = Bx, r) wih r Kρ ), 32) P B B ε ae in 4 B iii) For any ε, ) here exiss K > such ha, for any λ > and any ball B = Bx, r) wih r Kρ λ), 33) λr B λ B ε ae in 4 B

13 UPPER ESTIMATES 3 Remark 32 As one can see below, he doubling propery of ρ is mildly used only in he proof of he implicaion ii) iii) In fac, he doubling propery can be dropped from he hypoheses, bu hen condiions r Kρ ) and r Kρ λ) should be replaced respecively by r Kρ c) and r Kρ c λ), for a posiive consan c > Remark 33 If he hea semigroup P possesses he hea kernel p x, y) hen condiion i) can be equivalenly saed as follows: for any ε, ) here exiss K > such ha, for all >, r Kρ ), and almos all x M, 34) p x, y) dµ y) ε Bx,r) c Indeed, for any ball B x, r) and for almos all x B x, r/4) or even x B x, r/2)), we have P Bx,r) c x) = p x, y) dµ y) p x, y) dµ y), Bx,r) c Bx,r/2) c so ha 34) implies 3) wih K being replaced by 2K) Similarly, for almos all x B x, r/2), p x, y) dµ y) p x, y) dµ y) = P Bx,r/2) c x), Bx,r) c Bx,r/2) c so ha 3) implies 34), for almos all x B x, r/8) overing M by a counable family of balls of radius r/8, we obain ha 34) holds for almos all x M Proof of Theorem 3 i) ii) Applying esimae 429) of Lemma 48 o funcion f = 2 B, we obain ha 35) P B Bx) P Bx) sup essup P s 2 2 s [,] B, 3 4 B)c 2 for > and ae x M For any x 4 B, we have ha Bx, r/4) 2B see Fig 5) Using 3) and he ideniy P = ae, we obain, for any x 4 B, P 2 B = P 2 B)c P Bx,r/4) c Applying hypohesis i) for he ball B x, r/4), we obain ha P Bx,r/4) c ε ae in B x, r/6), provided r 36) Kρ ) 4 wih sufficienly large K I follows ha, for any x 4 B, whence P B ε ae in B x, r/6), 2 37) P 2 B ε ae in 4 B On he oher hand, for any y 3 4 B) c, we have 2 B B y, r/4)c see Fig 5), whence P s 2 B P s By,r/4) c Applying hypohesis i) for he ball B y, r/4) a ime s, we obain ha if 36) holds for sufficienly large K hen, for all < s, I follows ha, for any y 3 4 B) c, P s By,r/4) c ε ae in B y, r/6) P s B ε ae in B y, r/6), 2

14 4 GRIGOR YAN AND HU B=Bx,r) 3 /4 B /2 B Bx, /6 r) x /4 B x By, /6 r) Bx, /4 r) y By, /4 r) Figure 5 Illusraion o he proof of i) ii) whence 38) P s B ε ae in 2 ) 3 c 4 B ombining 35), 37) and 38), we obain ha, under condiion 36), 39) P B B P B 2 B 2ε ae in 4 B, which is equivalen o 32) ii) iii) By ii), we have 32) provided ρ /K), whence λr B λ B = λ e λ P B B d λ which holds almos everywhere in 4B If ρ r K ) e λ P B B d ε) e λρ r )) K, 3) λρ r K ) log ε hen we obain λr B λ B ε) 2 ae in 4 B, which is equivalen o 33) ondiion 3) is equivalen o ) log ε 3) r Kρ, λ which, by he doubling propery of ρ, is a consequence of r K ρ λ) for sufficienly large K iii) i) Le us firs show ha, for all, λ >, 32) P B B e λ λr B λ B) Indeed, using he facs ha P B s B B and P B s+ B = P B P B s B ) P B B,

15 we obain ha λr B λ B = λ = λ UPPER ESTIMATES 5 e λs P B s B ds e λs P B s B ds + λ e λ) + λ e λ + e λ P B B, e λs P B s B ds e λs+) P B s+ B ds hus giving 32) Given ε, ), > and r Kρ ) where K is defined by hypohesis iii)), choose λ from condiion r = Kρ λ) Then i follows from 33) and 32) ha P B P B B εe λ ae in 4 B Using he ideniy P = and observing ha λ λρ r ) =, K we obain P B c = P B εe λ εe ae in 4 B, which is equivalen o 3) The following saemen is an exension of Theorem 3 in he case of a local Dirichle form Theorem 34 Assume ha all he hypoheses of Theorem 3 hold, and in addiion ha he Dirichle form E, F) is local Then each of condiions i), ii), iii) of Theorem 3 is equivalen o he following: iv) There are c, > such ha, for any > and any ball B = Bx, r), 33) essup 2 B P B c exp where 34) Φ s) := sup λ> Φ c r { } s ρ /λ) λ Remark 35 Obviously, esimae 33) wih funcion Φ defined by 34) is equivalen o he following: for all λ >, 35) essup P B c exp λ cr ) 2 B ρ /λ) Remark 36 If he hea semigroup P possesses he hea kernel p x, y) hen condiion iv) can be equivalenly saed as follows: for all, r > and for almos all x M, 36) p x, y) dµ y) exp Φ c r )), Bx,r) c which is proved by he argumen of Remark 33 Esimae 35) can be reformulaed similarly Example 37 If ρ ) = /β for some β > hen so ha 33) becomes Φ s) = sup λ> { sλ /β λ essup 2 B P B c exp )), } = β s β r β c β ) β )

16 6 GRIGOR YAN AND HU Proof of Theorem 34 Observe firs ha he funcion Φ is non-negaive on [, + ) jus le λ in 34)), monoone increasing, and saisfies he following inequaliy: for all s and A, 37) Φ As) AΦ s), which is proved as follows: { As Φ As) = sup λ> A sup λ> = AΦ s) } ρ /λ) λ { } s ρ /λ) λ Le us prove ha iv) i) Assuming ha r 2c Kρ ) where K > is o be specified below) and using 37), 34), we obain Φ c r ) Φ 2K ρ ) ) ) 2ρ ) KΦ { } 2ρ ) = K sup λ> ρ /λ) λ K, where he las inequaliy follows by seing λ = / Hence, 33) implies ha P B c exp K) ae in 2 B hoosing K big enough, we obain 3) Now we prove he main implicaion iii) iv) This proof is raher long and will be spli ino five seps Sep We claim ha, for any ε >, here exiss K > such ha if a funcion w F L M) is such ha w in a ball B = B x, r) and w saisfies weakly in B he equaion w + λw =, where λ > and r are relaed by hen r Kρ ), λ w ε ae in 4 B Indeed, since he Dirichle form is local and he ball is precompac, we have by orollary 45 see Appendix) ha w λr B λ B ae in B By iii), we have λr B λ B ε ae in 4 B, provided r Kρ λ), where K is now defined by condiion iii) ombining he above wo lines, we finish he proof of he claim Sep 2 Le us show ha here exiss c > such ha, for any ball B = B x, r) and any λ >, 38) essup λr λ B c) exp cr ) Bx,δ) ρ /λ) +, where δ = δ λ) > hoose some R > 4r and consider he funcions φ = Bx,R)\Bx,r)

17 and 39) u = λr λ φ I suffices o prove ha 32) essup u exp cr ) Bx,δ) ρ /λ) +, UPPER ESTIMATES 7 and hen le R Since φ and φ L 2 M), we have u on M, u dom ) F, and u saisfies in M he equaion 32) u + λu = λφ I suffices o assume ha 322) cr ρ ), λ where c > is o be specified laer) because oherwise 32) is rivially saisfied due o u Le n 2 be an ineger o be deermined laer on For any i n, se r i = ir n, and, for i < n, b i = essup u, Bx,r i ) w i x) = ux) b i+ learly, w i F L M) Since φ = in B x, r), i follows from 32) ha w i + λw i = in B x, r) By definiion of b i+, we have w i in B x, r i+ ) In paricular, he same inequaliy holds in any ball B x, r ) for any x B x, r i ) see Fig 6) Therefore, by Sep wih ε = e, Bx,r) Bx,r i+) Bx,r i) Bx,r ) Bx, /4r ) x Bx, r ) Figure 6 Balls B x, r i ) and B x, r i+ ) we have ha w i e ae in B x, ) 4 r, provided 323) r Kρ ), λ

18 8 GRIGOR YAN AND HU for an appropriae consan K I follows ha ha is, essup w i e, Bx,r i ) 324) b i e b i+ Before we proceed furher, le us make sure ha condiion 323) is saisfied Since r = r/n, i is equivalen o r n Kρ ), λ so ha we can choose [ ] r n = Kρ ) λ hoosing in 322) c = 2K, we obain ha n 2 Noe also ha n 2cr ρ ) λ Now, ieraing 324) and using he fac ha b n, we obain ) b e n ) b n e n/2 exp cr ρ ) + λ learly, his implies 32), where δ can be anyhing r = r n ; for example, se δ = Kρ λ) Le us noe ha he ieraion argumen in his par of he proof is moivaed by ha in [4] for he seing of infinie graphs Sep 3 Le us show ha here is K such ha for any ball B = B x, r) wih 325) r Kρ we have λ ), 326) essinf 2B) c λr λ B c) 2 Indeed, for any x 2B) c, we have Bx, r) B c, whence by condiion iii), λr λ B c λr λ Bx,r) 2 ae in B x, ) 4 r provided 325) is saisfied wih an appropriae K Hence, 326) follows Sep 4 Le us show ha, for any non-negaive funcion f L M), he funcion u = λr λ f saisfies he inequaliy 327) P u e λ u in M for arbirary, λ > Indeed, we have P u = λ = λ = e λ λ e λs P +s f ds e λs ) P s f ds e λs P s f ds e λ u

19 UPPER ESTIMATES 9 Sep 5 Finally, le us prove 33) Le c be he same as in 38) Sep 2), so ha for any λ > and for u = λr λ B c, 328) essup u exp cr ) Bx,δ) ρ /λ) + Le λ > be such ha 325) is saisfied Then i follows from 326) ha u 2 2B) c in M Applying P o he boh sides of his inequaliy and using 327), we obain 329) which ogeher wih 328) yields 2 P 2B) c P u e λ u, 33) essup P 2B) c exp Bx,δ) λ cr ), ρ /λ) where = 2e If λ is such ha 325) fails, ha is, r < Kρ /λ) hen 33) holds rivially wih = e ck Hence, 33) holds for all λ >, which is equivalen o 35) In applicaions i is frequenly convenien o replace funcion Φ in 33) by a more explici funcion as in he following saemen Lemma 38 Define a funcion Ψλ) on [, + ) by { λρ ) 33) Ψ λ) = λ, λ >,, λ = and assume ha Ψ λ) is a coninuous, monoone increasing bijecion from [, + ) ono [, + ), so ha he inverse funcion Ψ is defined on [, ) Then 332) Φ 2s) Ψ s) Φ s) for all s Proof Se λ = Ψ s) so ha I follows from 34) ha Φ 2s) which proves he lef inequaliy in 332) Since s = λρ /λ) 2s ρ /λ) λ = λ, { λρ /λ) ρ /ν) ν Φ s) = sup ν> he righ inequaliy in 332) is equivalen o he inequaliy which afer division by ν becomes 333) λρ /λ) ρ /ν) }, ν λ, for all ν >, Ψ λ) Ψ ν) + λ ν Indeed, if ν λ hen by he monooniciy of Ψ, Ψ λ) Ψ ν), which obviously implies 333) If ν < λ hen ρ /ν) ρ /λ) and which implies 333) as well Ψ λ) λρ /λ) = Ψ ν) νρ /ν) λ ν,

20 2 GRIGOR YAN AND HU orollary 39 Theorem 34 remains rue if he funcion Φ in condiion iv) is replaced by Ψ, provided Ψ exiss 32 Poinwise esimaes of he hea kernel Now we can sae he main resul abou he relaion beween DUE) and UE) in he case of a local Dirichle form Le us firs sae and label all he required condiions in erms of he funcions V and ρ: The upper bounds for he volume of balls: for all x M and r > V ) µ Bx, r)) V r) The version of condiion T ): for any ε > here is K > such ha whenever r Kρ ) hen, for almos all x M, T weak ) p x, z) dµz) ε Bx,r) c Obviously, T weak ) is equivalen o he fac ha T ) holds wih some funcion h such ha h s) as s The on-diagonal upper bound: for all > and almos all x, y M, DUE) p x, y) V ρ )), UE exp ) The upper bound wih he exponenial ail: for all > and almos all x, y M, )) p x, y) Φ c, where Φ is defined by 34) V ρ)) exp dx, y) Theorem 3 Le E, F) be a regular, conservaive, local Dirichle form in L 2 M) and le p x, y) be is hea kernel Assume ha all meric balls in M are precompac and saisfy V ), and le funcions ρ and V be doubling Then Proof Le us prove he implicaion DUE) + T weak ) UE exp ) DUE) + T weak ) UE exp ) Observe firs ha condiion T weak ) is equivalen o he condiion i) of Theorem 3 cf Remark 33) Hence, by Theorem 34, we obain 334) p x, z)dµ z) exp Φ c r )), Bx,r) c for all, r > and almos all x M cf Remark 36) Using he semigroup propery, DUE), and 334), we obain, for almos all x, y M and seing r = 2d x, y), ha p 2 x, y) = p x, z)p y, z)dµ z) M p x, z)p y, z) dµz) + p x, z)p y, z) dµz) Bx,r) c By,r) c essup p, ) p x, z) dµz) + essup p, ) p y, z) dµz) M M Bx,r) c M M By,r) c V ρ )) exp Φ c r )) Renaming 2 by and applying 37) and he doubling propery of V and ρ, we obain UE exp ) Noe ha in his par of he proof we have no used V ) Le us now prove he converse, ha is, UE exp ) DUE) + T weak )

21 UPPER ESTIMATES 2 The on-diagonal bound DUE) follows from UE exp ) rivially To prove T weak ), observe ha, by UE exp ), Bx,r) c p x, z) dµz) exp Φ V ρ)) Bx,r) c c dx, z) Hence, we are lef o prove ha, for any ε > here is K = K ε) such ha )) cdx, z) 335) exp Φ dµz) ε, V ρ)) Bx,r) c provided 336) r Kρ ) For any non-negaive ineger k, se ) 337) ξ k = Φ c 2k r and observe ha, by 37), 338) ξ k 2 k ξ )) dµz) Nex, consider he following par of he inegral 335): )) cdx, z) I k = exp Φ dµz) V ρ)) Bx,2 k+ r)\bx,2 k r) V 2 k+ r ) c2 k )) r exp Φ V ρ)) 2 k+ ) α r exp ξ ρ ) k ) ) r α ξ α 339) ρ ) ξ k exp ξ k), where we have used V ), 2) which is a consequence of he doubling propery of V ), 337), and 338) Observe ha by 337) and 34) ξ = Φ c r ) { } cr = sup λ> ρ /λ) λ cr ρ ), which follows by seing λ = / Assuming ha r and are relaed by 336) and K is so large ha 34) ck 2, we obain 34) ξ c r 2 ρ ) Subsiuing ino 339), we obain 342) I k ξ α k exp ξ k)

22 22 GRIGOR YAN AND HU On he oher hand, using ξ k /ξ k 2, which is rue by 37), we obain ξk s α exp s) ds = s α exp s) ds ξ ξ k where c = α 2 α ) I follows ha k= k= exp ξ k ) k= c k= ξ α k exp ξ k) = ξ α exp ξ ) + ξk ξ α k exp ξ k), ξ α exp ξ ) + exp ξ ) 2 k= Therefore, we obain from 342), 34), 336) 343) I k exp ξ ) exp c ) r 2 4 ρ ) k= ξ k s α ds ξ α k exp ξ k) ξ s α exp s) ds exp ck 4 Since he lef hand side of 335) is equal o k= I k and K can be chosen arbirarily large, we obain ha 335) can be saisfied wih any ε >, which finishes he proof Using Lemma 38, we obain immediaely he following consequence orollary 3 Theorem 3 remains rue if he funcion Φ in UE exp ) is replaced by Ψ, provided Ψ exiss I follows from 343) ha, for all, r > and almos all x M, 344) p x, z) dµz) exp c ) r Bx,r) c 4 ρ ) Indeed, 343) was proved above under he assumpion ha r/ρ ) 2/c cf 336) and 34)) If r/ρ ) < 2/c hen 344) is rivially saisfied wih = e /2 Hence, under he hypoheses of Theorem 3, we obain T ) wih he funcion 345) h s) = exp cs) Le us inroduce a sronger version of condiion T ), wih an addiional requiremen on h: for all, r > and almos all x M, ) r p x, z) dµz) h, T srong ) Bx,r) c ) ρ) r and lim h = ρ) For example, funcion 345) saisfies he second condiion in T srong ) if, for some c, η >, 346) ρ ) c η, < < The above observaion means ha, under he hypoheses of Theorem 3 and assuming in addiion 346), he following is rue: UE exp ) DUE) + T srong ) )

23 The converse implicaion UPPER ESTIMATES 23 DUE) + T srong ) UE exp ) is rue as well, jus because T srong ) implies T weak ) Hence, he hypohesis T weak ) in he saemen of Theorem 3 can be replaced by T srong ), provided one assumes in addiion 346) Besides, he condiion T srong ) allows o drop he hypohesis of he localiy of he Dirichle form in Theorem 3 as i is clear from he following saemen Lemma 32 Le E, F) be a conservaive Dirichle form wih he hea kernel p x, y) T srong ) holds hen he Dirichle form E, F) is local If Proof We need o prove ha if f and g are wo funcions from F whose suppors are disjoin compac ses hen E f, g) = Since E, F) is conservaive, we have E f, g) = lim E f, g), where E is defined by 8) Hence, i suffices o show ha 347) E f, g) as Le A = supp f and = supp g Since A and are disjoin, i follows from 8) ha E f, g) = f x) g y) p x, y) dµ y) dµ x), whence by he auchy-schwarz inequaliy E f, g) A A f g A essup y ) /2 f 2 x) p x, y) dµ y) dµ x) ) /2 g 2 y) p x, y) dµ y) dµ x) essup x A A ) /2 p x, y) dµ y) p x, y) dµ x)) /2 Le r = dis A, ) hoose a finie covering {B i } N i= of A by meric balls B i = B x i, r/2) x i A) Then 2B i ) c for any i, and we obain by T srong ) essup p x, y) dµ y) x A sup essup p x, y) dµ y) i x B i 2B i ) c ) 2r h as ρ ) Similarly, we have ha whence 347) follows essup p x, y) dµ x) as, y A ombining Lemma 32 wih he previous remarks, we obain he following resul orollary 33 Under he hypoheses of Theorem 3, drop he assumpion of he localiy of E, F) and add condiion 346) Then he following is rue: DUE) + T srong ) UE exp )

24 24 GRIGOR YAN AND HU 4 Appendix: Markovian properies We prove here a number of he consequences of he Markov propery of he Dirichle forms, such as he maximum and comparison principles, he properies of he resolvens and he hea semigroups in subses, ec, which are necessary for Secion 3 orollary 45 and Lemma 48 are explicily used in he proofs of Theorem 34 and 3, respecively) These resuls are wellknown, bu i is hardly possible o give accurae references Besides, he exising proofs would normally use he Hun process associaed wih he Dirichle form We give self-conained, analyic proofs of all hese resuls, some of which are new Le us sae some frequenly used basic facs abou Dirichle forms: Exension of he Markov propery) If ϕ : R R is a Lipschiz funcion wih he Lipschiz consan and ϕ ) = hen, for any u F, he funcion ϕ u) is also in F and E ϕ u)) E u) For he funcion ϕ s) = maxmin s, ), ) his propery holds by he definiion of he Markov propery The proof for a general ϕ can be found in [, Theorem 4, p23] If u, v F L M) hen uv F see [, Theorem 42ii)]) Definiion 4 For any open subse Ω of M, le F Ω) be he se of funcions from F whose suppor is compac and is conained in Ω Then define F Ω) as he closure of F Ω) in F wih respec o he E -norm In paricular, i follows ha any funcion from F Ω) vanishes in Ω c and, hence, can be idenified as an elemen of L 2 Ω) If F Ω) is dense in L 2 Ω) hen E, F Ω)) is a Dirichle form in L 2 Ω) In his case, denoe by Ω and P Ω respecively he generaor and he semigroup of E, F Ω)) If f L 2 M) hen se P Ω f := P Ω f Ω ) In general F Ω) need no be dense in L 2 Ω) However, if he Dirichle form E, F) is regular hen F Ω) is obviously dense in L 2 Ω) In his case, F Ω) admis he following wo equivalen definiions see [, orollary 23, p95 and Theorem 442, p54]): ) F Ω) is he closure of F Ω) in F 2) F Ω) = {f F : f } = qe on Ω c where f is a quasi-coninuous modificaion of f and qe sands for quasi everywhere Le us sae he following useful properies of regular Dirichle forms: If E, F) is a regular Dirichle form hen E, F Ω)) is also a regular Dirichle form For any open se Ω M and any se S Ω, here is a funcion ϕ F Ω) such ha ϕ and ϕ in an open neighborhood of S see [, p27]) Such a funcion ϕ is called a cu-off funcion of he pair S, Ω) 4 Maximum principle for weak soluions In his subsecion, E, F) is a Dirichle form in L 2 M), no necessarily regular, unless oherwise saed Lemma 42 []) Le u, v be wo funcions from F such ha u and v If u on he se {v > } hen E u, v) Proof I follows from he hypoheses ha, for any λ >, min u + λv, ) = u By he Markov propery and he bilineariy of he Dirichle form, we obain Eu) Eu + λv) = Eu) + 2λEu, v) + λ 2 Ev), whence 2λEu, v) + λ 2 Ev) Dividing by λ and hen leing λ, we obain E u, v), which was o be proved

25 UPPER ESTIMATES 25 Lemma 43 Le ϕ s) be an increasing funcion on R such ha ϕ ) = and ϕ s ) ϕ s 2 ) s s 2 for all s, s 2 R Then, for any u F, also ϕ u) F and 4) E u, ϕ u)) E ϕ u)) In paricular, 4) implies 42) E u, ϕ u)) For example, applying his wih he funcion ϕ s) = s + := max s, ), we obain ha 43) Eu, u + ) Proof Tha ϕ u) belongs o F is rue by he Markov propery Fix some λ, ) and consider he funcion 44) ψ s) = λs + λ) ϕ s) Obviously, ψ is Lipschiz and ψ ) =, whence i follows ha ψ u) F Using ϕ where all relaions involving he derivaives of Lipschiz funcions are undersood almos everywhere), we obain from 44) 45) ψ max λ, ϕ ) In paricular, ψ λ, which implies ha he funcion ψ has he inverse ψ on R, which is also a Lipschiz funcion Using he ideniy ϕ ψ ) s) = ϕ ψ s) ) ψ ψ s) ) and ψ ϕ cf 45)), we obain which implies by he Markov propery ha ϕ ψ ), 46) E ϕ u)) = E ϕ ψ ) ψ u)) ) E ψ u)) On he oher hand, by 44), E ψ u)) = λ 2 E u) + λ) 2 E ϕ u)) + 2λ λ) E u, ϕ u)) Expanding in λ and comparing wih 46), we obain 2λ E u, ϕ u)) E ϕ u))) + λ 2 E u) 2E u, ϕ u)) + Eϕ, u)) Dividing by λ and hen leing λ, we obain 4) The maximum principles ha will be saed below in Proposiions 46 and 4, use he boundary condiion 47) u on Ω c, ha is o be undersood in a weak sense The precise meaning of 47) is ha u + F Ω) The nex saemen provides a convenien equivalen way of saing his condiion Lemma 44 Le E, F) be a regular Dirichle form Le u F and Ω be an open subse of M Then he following are equivalen: i) u + F Ω) ii) u v in M for some funcion v F Ω)

26 26 GRIGOR YAN AND HU Proof The implicaion i) ii) is rivial since we can ake v = u + Le us prove ii) i) Se f = u v so ha f F and f in M The quesion amouns o proving ha 48) v + f) + F Ω), for any non-posiive f F and any v F Ω) Assume firs ha f F L M) and v F Ω) Le ϕ be a cu-off funcion of supp v in Ω Since ϕ F L M) and supp ϕ Ω, i follows ha ϕf F Ω) Observe ha 49) v + f) + = v + ϕf) + Indeed, on supp v we have ϕ so ha he ideniy 49) is rivially saisfied, while on he se {v = } he boh sides of 49) vanish because f see Fig 7) Since v + ϕf F Ω), we Figure 7 Funcion v + ϕf conclude ha v + ϕf) + F Ω) whence 48) follows For an arbirary non-posiive funcion f F, consider he sequence f k = max f, k) so ha f k F L E M), f k, and f k f see [, Theorem 42iii), p26]) Also, if v F Ω) hen here is a sequence of funcions {v k } k= F E Ω) such ha v k v By he previous argumen, we have 4) v k + f k ) + F Ω) Since i follows by [, Theorem 42v), p26] ha v k + f E v + f, E v k + f) + v + f)+ = u +, where he convergence is weak wih respec o E -norm However, F Ω) being a closed subspace of he Hilber space F, is also weakly closed Togeher wih 4), his implies u + F Ω), which was o be proved In he wo proposiions below, he regulariy of E, F) is no assumed Definiion 45 Le Ω be an open subse of M, f L 2 Ω) and λ R We say ha a funcion u F saisfies weakly he inequaliy if, for any non-negaive funcion ψ FΩ), u + λu f in Ω, 4) Eu, ψ) + λ u, ψ) f, ψ) Similarly one defines in he weak sense he inequaliy u + λu f and he ideniy u + λu = f

27 UPPER ESTIMATES 27 Proposiion 46 Ellipic maximum principle) Le u F be a funcion such ha, for some open se Ω M and λ >, { u + λu weakly in Ω, Then u ae in Ω u + F Ω) Proof Since u + FΩ), we can ake ψ = u + in 4) and obain ha Eu, u + ) + λ u, u + ) Since λ > and by 43) Eu, u + ), i follows ha whence u + = in Ω u + L 2 Ω) = u, u +), Remark 47 By Lemma 44, in he case when he Dirichle form is regular, condiion u + F Ω) can be replaced by u v for some v F Ω) Remark 48 For he case λ = some addiional assumpions on he domain Ω are necessary for he validiy of he maximum principle Under differen assumpions on Ω and for a local form E, he following version of he ellipic maximum principle was proved in [5, p4] and [7, orollary ]: if u F, u weakly in Ω and ũ is a quasi-coninuous version of u hen ũ qein M \ Ω implies ha ũ qe in Ω, where is an arbirary consan Noe ha he condiion ũ qe in M \ Ω is equivalen o u + F Ω) cf he remark afer Definiion 4) Definiion 49 Le I be an open inerval in R, Ω be an open subse of M, and f L 2 Ω) We say ha a funcion u : I F saisfies weakly he inequaliy 42) u u f in I Ω, if he Fréche derivaive u of u exiss in L2 Ω) for any I and, for any non-negaive funcion ψ F Ω), ) u 43), ), ψ + E u, ), ψ) f, ψ) Similarly one defines in he weak sense he inequaliy u u u f and he ideniy u = f u = weakly in I Ω hen he funcion u is called a weak soluion o he hea If u equaion in I Ω The weak inequaliy u u ) defines a weak subsoluion resp, supersoluion) o he hea equaion Example 4 For example, for any f L 2 M), he funcion u = P f is a weak soluion o he hea equaion in, + ) M, ha is, for any ψ F M), ) 44) P f, ψ + E P f, ψ) = Indeed, i follows from he specral heory ha he following equaion is saisfied srongly P f = P f), ha is, is undersood here as an operaor in L 2 M) By he definiion of he generaor, we have P f), ψ) = E P f, ψ), whence 44) follows

28 28 GRIGOR YAN AND HU Proposiion 4 parabolic maximum principle) Fix T, + ] and an open subse Ω M, and assume ha a funcion u :, T ) F saisfies he following condiions: Then u ae on, T ) Ω u u weakly in, T ) Ω, u +, ) FΩ) for any, T ) u +, ) L2 Ω) as Proof Fix a funcion ϕ R) such ha ϕ = on, ], ϕ > on, ), and ϕ on R hoosing in 43) he funcion we obain ψ := ϕu, )) = ϕu +, )) FΩ), ) u, ϕu) + E u, ϕu)) By 42), we have E u, ϕu)), whence ) u 45), ϕu) Now define he funcion Φ by s /2 Φs) = ϕξ) dξ), s R By choosing a suiable ϕ in a righ neighborhood of, for example, leing ϕs) = d ds exp s 2), we can make Φ s) and all is derivaives Φ k) s) end o as s, so ha Φ R) Noe ha Φ = on, ], Φ > on, ), and Φ on R All hese properies are obvious excep ha Φ on, ) The laer is proved as follows: since we have d ds s ϕξ) dξ = ϕs) ϕ s)ϕs) = d 2 ds ϕ2 s), Φs) 2 = s ϕξ) dξ 2 ϕ2 s), s >, whence Φ s) = ϕs) 2Φs) 2 2Φs) = 2Φs) 2 < I follows ha Φ u) F I is easy o show ha he funcion Φ u, )) is Fréche differeniable in L 2 Ω) and, by he chain rule, Φ u) = Φ u) u By he produc rule for he Fréche derivaive, we obain from 45) ha d Φu), Φu)) = 2 Φ u) u ) d, Φu) ) ) u u =, 2Φ u)φu) =, ϕu) Hence, he funcion Φu, )) is non-increasing in As Φs) s +, i follows ha, for any, T ), Φu, )) lim Φus, )) lim u +s, ) =, s + s + which implies ha u ae on, T ) Ω

29 UPPER ESTIMATES omparison lemmas for he resolven In his subsecion, E, F) is a regular Dirichle form in L 2 M) For any open subse Ω M and any λ >, define he resolven operaor Rλ Ω : L2 Ω) L 2 Ω) by 46) R Ω λ f = Ω + λ) f = e λ P Ω f d, for any f L 2 Ω) I follows ha R Ω λ is a bounded operaor in L2 Ω) and, for any f L 2 Ω), R Ω λ f dom Ω) F Ω) I is clear from 46) ha f implies R Ω λ f and f implies λrω λ f If f L 2 M) hen se R Ω λ f := RΩ λ f Ω) Lemma 42 Le Ω M be an open se, λ >, and le a non-negaive funcion u F saisfy weakly in Ω he inequaliy 47) u + λu f, where f L 2 Ω) Then u R Ω λ f Proof I follows from he definiion 46) of he resolven ha he funcion v = R Ω λ f saisfies in Ω he equaion 48) Ω v + λv = f Muliplying 48) by ψ F Ω) and inegraing over Ω, we obain ha is, v saisfies weakly he equaion E v, ψ) + λ v, ψ) = f, ψ), v + λv = f in Ω I follows from 47) ha he funcion w = v u belongs o F and saisfies weakly he inequaliy w + λw in Ω Since w v and v F Ω), we conclude by Lemma 44 ha w + F Ω) Then by Proposiion 46, we obain w in Ω, ha is, u v, which was o be proved I follows from Lemma 42 ha he funcion u = Rλ Ω f is he minimal non-negaive soluion from he class F o he equaion u + λu = f in Ω, which is undersood in he weak sense Lemma 43 If {Ω i } i= is an increasing sequence of open subses of M and Ω = i= Ω i hen, for any λ > and any f L 2 Ω), R Ω i λ f ae R Ω λ f Proof Se u i = R Ω i α f and observe ha, by Lemma 42, he sequence {u i } is increasing and u i R Ω αf Therefore, u i converges almos everywhere o a measurable funcion u on Ω such ha u R Ω αf This implies ha u L 2 Ω) and, by he dominaed convergence heorem, u i u in L 2 Ω) We need o prove ha u = R Ω λ f Le us firs show ha u F Ω) The funcion u i belongs F Ω i ) and, hence, u i F Ω) Le us show ha he sequence {u i } is auchy in F Ω) wih respec o he norm E Each funcion u i saisfies he equaion 49) E u i, ϕ) + α u i, ϕ) = f, ϕ),

30 3 GRIGOR YAN AND HU for any ϕ F Ω i ) hoosing here ϕ = u i, we obain E u i, u i ) + α u i, u i ) = f, u i ) Fix k > i and observe ha he funcion ϕ = u k 2u i belongs o F Ω k ) Therefore, by he analogous equaion for u k, we obain Adding up he above wo lines yields whence E u k, u k 2u i ) + α u k, u k 2u i ) = f, u k 2u i ) E u k ) + E u i ) 2E u k, u i ) + α u k 2 + u i 2 2 u k, u i ) ) = f, u k u i ), E u k u i ) + α u k u i 2 = f, u k u i ) f u k u i Since u k u i as k, i, we conclude ha also E u k u i ) and, hence, E u k u i ) Therefore, he sequence {u i } is auchy in F Ω) and, hence, converges in F Ω) Since is limi in L 2 Ω) is u, we conclude ha he limi of {u i } in F Ω) is also u In paricular, u F Ω) Now we can show ha u = R Ω αf Fix a funcion ϕ F Ω) and observe ha he suppor of ϕ is conained in Ω i when i is large enough Therefore, 49) holds for his ϕ for all large enough i Passing o he limi as i, we obain ha he same equaion holds for u insead of u i, ha is, 42) E u, ϕ) + α u, ϕ) = f, ϕ) Since F Ω) is dense in F Ω), his ideniy holds for all ϕ F Ω) Since he funcion R Ω αf belongs o F Ω) and also saisfies 42), we obain ha he funcion v = u R α f belongs o F Ω) and saisfies he ideniy E v, ϕ) + α v, ϕ) =, for all ϕ F Ω) Seing ϕ = v, we obain v =, which finishes he proof The following saemen is a modificaion of Lemma 42 in he case of a local form, where he hypoheses u in M can be relaxed o u in Ω Lemma 44 Le he Dirichle form E, F) be local Le Ω M be an open se, λ >, and le a funcion u F L M) be non-negaive in Ω and saisfy weakly in Ω he inequaliy 42) u + λu f, where f L 2 Ω) Then u R Ω λ f Proof I suffices o prove ha u R U λ f, for any open se U Ω and hen ake an exhausion of Ω by such ses U and pass o he limi by Lemma 43 Le ϕ be a cu-off funcion of he pair U, Ω) Then ϕu F and, since ϕu is suppored in Ω, i follows ha ϕu F Ω) Observe ha ϕu Le us apply Lemma 42 o he funcion ϕu insead of u and in he space Ω insead of M For ha, we need o verify ha he following inequaliy holds weakly in U: Indeed, for any ψ F U), we have where we have used ϕu) + λ ϕu) f E ϕu, ψ) + λ ϕu, ψ) = E ϕ ) u, ψ) + E u, ψ) + λ u, ψ) f, ψ), E ϕ ) u, ψ) =, which is rue by he localiy of he form E, F), and E u, ψ) + λ u, ψ) f, ψ),

31 which is rue by 42) By Lemma 42, we conclude ha in U which was o be proved UPPER ESTIMATES 3 u = ϕu R U λ f, orollary 45 Assume ha he Dirichle form E, F) is local Le Ω M be a precompac open se and λ > If a funcion w F L M) is such ha w in Ω and w saisfies weakly in Ω he inequaliy 422) w + λw, hen 423) w λr Ω λ Ω in Ω Proof Le ϕ be a cu-off funcion of he pair Ω, M) and consider he funcion u = ϕ w see Fig 8) learly, u F L M) and u in Ω Le us show ha u saisfies weakly in Ω he R λ Figure 8 Funcions w, ϕ, R Ω λ Ω inequaliy u + λu λ Indeed, for any ψ F Ω), we have E u, ψ) + λ u, ψ) = E ϕ, ψ) + λ ϕ, ψ) E w, ψ) + λ w, ψ)) λ, ψ), where we have used ha E ϕ, ψ) by Lemma 42, ϕ, ψ) =, ψ), and E w, ψ) + λ w, ψ) by 422) By Lemma 44, we conclude ha u λrλ Ω Ω, whence i follows ha in Ω w = ϕ u = u λrλ Ω Ω, proving 423) 43 omparisons lemmas for he hea semigroup In his subsecion, E, F) is a regular Dirichle form in L 2 M) Lemma 46 Le U be an open subse of M, and f L 2 U) If u : R + F is a weak non-negaive supersoluion o he hea equaion in R + U and 424) u, ) L2 U) f as hen, for all >, 425) u, ) P U f

32 32 GRIGOR YAN AND HU Proof Funcion P U f is a weak soluion o he hea equaion in R + U cf Example 4), and saisfies he iniial condiion 424) Hence, for he difference w = P U f u, we have w w weakly in R + U, w +, ) F U) for any >, w, ) L2 U) as, where he middle condiion follows from w, ) P U f F U) and Lemma 44 By Proposiion 4, we conclude ha w, whence 425) follows In paricular, if Ω is an open se conaining U hen applying Lemma 46 o u = P Ω f we obain P Ω f P U f Lemma 47 If {Ω i } i= is an increasing sequence of open subses of M and Ω = i= Ω i hen, for any > and any f L 2 Ω), 426) P Ω i f ae P Ω f as i Proof Assume firs ha f L 2 Ω ) The sequence of funcions {P Ω i f} i= is increasing and is bounded by P Ω f Hence, for any >, he sequence {P Ω i f} converges almos everywhere o a measurable funcion u on Ω such ha u P Ω f We need o show ha u = P Ω f Since u L 2 Ω), he dominaed convergence heorem implies ha P Ω i f L2 Ω) u Since he semigroup { P Ω } is srongly coninuous, he funcion P Ω f is coninuous as a pah in L 2 Ω), for all Le us prove ha he pah u is coninuous in L 2 Ω) For all s > and, we have P Ω i +s f P Ω i f = Since P Ω s f P Ω i s Leing i, we obain P Ω i P Ω i s f f ) P Ω i s f f f Ps Ω f, i follows ha P Ω i +s f P Ω i f P Ω s f f + P Ω s f f u +s u P Ω s f f + P Ω s f f as s, which means ha u is righ coninuous If > s > hen we have P Ω i s f P Ω i f = P Ω i s f P Ω i s f ) P Ω i s f f Arguing as above, we obain ha u is also lef coninuous Fix a non-negaive funcion ϕ F Ω) and observe ha ϕ F Ω i ) for large enough i I follows from 46) and he monoone convergence heorem ha, for any α >, R Ω i α f, ϕ) = e α P Ω i f, ϕ)d as i On he oher hand, by Lemma 43, whence i follows ha R Ω i α f, ϕ) R Ω αf, ϕ ) = e α u, ϕ) d = e α u, ϕ) d, e α P Ω f, ϕ ) d, e α P Ω f, ϕ ) d

33 UPPER ESTIMATES 33 Since u, ϕ) P Ω f, ϕ ) and he funcions u, ϕ), P Ω f, ϕ ) are coninuous in, his ideniy is only possible when 427) u, ϕ) = P Ω f, ϕ ) for all > I follows ha u = P Ω f, which was claimed Finally, consider an arbirary non-negaive funcion f L 2 Ω) Fix k N and se f k = f Ωk By he previous par of he proof, we have P Ω L i f 2 Ω) k P Ω f k as i For any i > k, we have P Ω f P Ω i f P Ω f P Ω f k + P Ω f k P Ω i 2 f f k + P Ω f k P Ω i f k f k + P Ω i f k P Ω i f whence i follows ha lim sup i Leing k, we obain lim i P Ω f P Ω i P Ω f P Ω i f 2 f f k f =, whence 426) follows Lemma 48 For any wo open subses U Ω of M, for any compac se K U, for any f L 2 M) and all >, 428) essup P Ω f P U f ) sup essup Ps Ω f Ω s [,] Ω\K In paricular, applying 428) for Ω = M, we obain, for any f L 2 M), > and almos all x M, 429) P fx) P U fx) sup essup P s f s [,] K c Proof Le {Ω i } i= be an increasing sequence of precompac open ses ha exhauss Ω and {U i } i= be a similar sequence o exhaus U and such ha K U i Ω i for all i By 426) and P U i f P U f, i suffices o prove ha ) essup P Ω i f P U i f sup essup P Ω i s f Ω i s [,] Ω i \K Hence, renaming Ω i o Ω and U i o U, we can assume in he sequel ha U and Ω are precompac Fix some T > and se m = sup s [,T ] essup Ps Ω f Ω\K If m hen 428) is rivially saisfied for = T Assuming in he sequel ha m <, choose a cu-off funcion ϕ of he couple Ω, M), and consider he funcion 43) u = P Ω f P U f mϕ I suffices o prove ha u T in Ω In fac, we shall prove ha u in Ω for all [, T ] For any [, T ], we have u F and u in M \ K, he laer being rue by he definiion of m I follows ha u ) + = in M \ K and, hence, u ) + FU) By Proposiion 4, in order o prove ha u in U, i suffices o verify ha u is a weak subsoluion o he hea