Elliptic Harnack inequalities for symmetric non-local Dirichlet forms

Transcription

1 Elliptic Harnack inequalities for symmetric non-local Dirichlet forms Zhen-Qing Chen, Takashi Kumagai and Jian Wang Abstract We study relations and characterizations of various elliptic Harnack inequalities for symmetric non-local Dirichlet forms on metric measure spaces. We allow the scaling function be state-dependent and the state space possibly disconnected. Stability of elliptic Harnack inequalities is established under certain regularity conditions and implication for a priori Hölder regularity of harmonic functions is explored. New equivalent statements for parabolic Harnack inequalities of non-local Dirichlet forms are obtained in terms of elliptic Harnack inequalities. Introduction and Main Results The classical elliptic Harnack inequality asserts that there exists a universal constant c = c d) such that for every x R d, r > 0 and every non-negative harmonic function h in the ball Bx 0, 2r) R d, ess sup Bx0,r)h c ess inf Bx0,r)h..) A celebrated theorem of Moser [M]) says that such elliptic Harnack inequality holds for nonnegative harmonic functions of any uniformly elliptic divergence operator on R d. One of the important consequences of Moser s elliptic Harnack inequality is that it implies a priori elliptic Hölder regularity see Definition.0 below) for harmonic functions of uniformly elliptic operators of divergence form. Because of the fundamental importance role played by a priori elliptic Hölder regularity for solutions of elliptic and parabolic differential equations, elliptic Harnack inequality and parabolic Harnack inequality, which is a parabolic version of the Harnack inequality see Definition.7 below), have been investigated extensively for local operators diffusions) on various spaces such as manifolds, graphs and metric measure spaces. It is also very important to consider whether such Harnack inequalities are stable under perturbations of the associated quadratic forms and under rough isometries. The stability problem of elliptic Harnack inequality is a difficult one. In [B], R. Bass proved stability of elliptic Harnack inequality under some strong global bounded geometry condition. Quite recently, this assumption has been relaxed significantly by Barlow-Murugan [BM]) to bounded geometry condition. Research partially supported by the Grant-in-Aid for Scientific Research A) and 7H0093, Japan. Research partially supported by the National Natural Science Foundation of China No ), the JSPS postdoctoral fellowship ), the Fok Ying Tung Education Foundation No. 5002), National Science Foundation of Fujian Province No. 205J0003), the Program for Probability and Statistics: Theory and Application No. IRTL704), and Fujian Provincial Key Laboratory of Mathematical Analysis and its Applications FJKLMAA).

2 For non-local operators, or equivalently, for discontinuous Markov processes, harmonic functions are required to be non-negative on the whole space in the formulation of Harnack inequalities due to the jumps from the processes; see EHI elliptic Harnack inequality) in Definition.2i) below. In Bass and Levin [BL], this version of EHI has been established for a class of nonlocal operators. If we only require harmonic functions to be non-negative in the ball Bx 0, 2r), then the classical elliptic Harnack inequality.) does not need to hold. Indeed, Kassmann [K] constructed such a counterexample for fractional Laplacian on R d. On the other hand, for nonlocal operators, it is not known whether EHI implies a priori elliptic Hölder regularity EHR) and we suspect it is not although parabolic Harnack inequality PHI) does imply parabolic Hölder regularity and hence EHR, see Theorem.9 below). To address this problem, some versions of elliptic Harnack inequalities that imply EHR are considered in some literatures such as [CKP, CKP2, K], in connection with the Moser s iteration method. We note that there are now many related work on EHI and EHR for harmonic functions of non-local operators; in addition to the papers mentioned above; see, for instance, [BS, CK, CK2, CS, DK, GHH, Ha, HN, K2, Sil] and references therein. This is only a partial list of the vast literature on the subject. The aim of this paper is to investigate relations among various elliptic Harnack inequalities and to study their stability for symmetric non-local Dirichlet forms under a general setting of metric measure spaces. This paper can be regarded as a continuation of [CKW, CKW2], which are concerned with the stability of two-sided heat kernel estimates, upper bound heat kernel estimates and PHI for non-local Dirichlet forms on general metric measure spaces. We point out that the setting of this paper is much more general than that of [CKW, CKW2] in the sense that the scale function in this paper can be state-dependent. As a byproduct, we obtain new equivalent statements for PHI in terms of EHI.. Elliptic Harnack inequalities Let M, d) be a locally compact separable metric space, and µ a positive Radon measure on M with full support. We will refer to such a triple M, d, µ) as a metric measure space. Throughout the paper, we assume that µm) =. We emphasize that we do not assume M to be connected nor M, d) to be geodesic. We consider a regular symmetric Dirichlet form E, F) on L 2 M; µ) of pure jump type; that is, Ef, g) = fx) fy))gx) gy)) Jdx, dy), f, g F, M M\diag where diag denotes the diagonal set {x, x) : x M} and J, ) is a symmetric Radon measure on M M \ diag. Since E, F) is regular, each function f F admits a quasi-continuous version f see [FOT, Theorem 2..3]). In the paper, we will always represent f F by its quasi-continuous version without writing f. Let L be the negative definite) L 2 -generator of E; this is, L is the self-adjoint operator in L 2 M; µ) such that Ef, g) = Lf, g for all f DL) and g F, where, denotes the inner product in L 2 M; µ). Let {P t } t 0 be its associated L 2 -semigroup. Associated with the regular Dirichlet form E, F) on L 2 M; µ) is an µ-symmetric Hunt process X = {X t, t 0; P x, x M \ N }. Here N is a properly exceptional set for E, F) in the sense that µn ) = 0 and P x X t N for some t > 0) = 0 for all x M \ N. This Hunt process is unique up to a properly exceptional set see [FOT, Theorem 4.2.8]. We fix X and N, and write M 0 = M \ N. While the semigroup {P t, t 0} associated with E is defined on L 2 M; µ), a 2

3 more precise version with better regularity properties can be obtained, if we set, for any bounded Borel measurable function f on M, P t fx) = E x fx t ), x M 0. Definition.. Denote by Bx, r) the ball in M, d) centered at x with radius r, and set V x, r) = µbx, r)). i) We say that M, d, µ) satisfies the volume doubling property VD) if there exists a constant C µ such that for all x M and r > 0, V x, 2r) C µ V x, r)..2) ii) We say that M, d, µ) satisfies the reverse volume doubling property RVD) if there exist positive constants d and c µ such that for all x M and 0 < r R, V x, R) R ) d. µ V x, r) c.3) r Since µ has full support on M, we have V x, r) = µbx, r)) > 0 for every x M and r > 0. The VD condition.2) is equivalent to the existence of d 2 > 0 and C µ so that V x, R) V x, r) C R ) d2 µ for all x M and 0 < r R..4) r The RVD condition.3) is equivalent to the existence of constants l µ > and c µ > so that V x, l µ r) c µ V x, r) for every x M and r > 0. It is known that VD implies RVD if M is connected and unbounded. In fact, it also holds that if M is connected and.2) holds for all x M and r 0, R 0 ] with some R 0 > 0, then.3) holds for all x M and 0 < r R R 0. See, for example [GH, Proposition 5. and Corollary 5.3]. Let R + := [0, ) and φ : M R + R + be a strictly increasing continuous function for every fixed x M with φx, 0) = 0 and φx, ) = for all x M, and satisfying that i) there exist constants c, c 2 > 0 and β 2 β > 0 such that c R r ) β φx, R) φx, r) c 2 R r ) β2 for all x M and 0 < r R;.5) ii) there exists a constant c 3 such that φy, r) c 3 φx, r) for all x, y M with dx, y) r..6) Recall that a set A M is said to be nearly Borel measurable if for any probability measure µ 0 on M, there are Borel measurable subsets A, A 2 of M so that A A A 2 and that P µ 0 X t A 2 \ A for some t 0) = 0. The collection of all nearly Borel measurable subsets of M forms a σ-field, which is called nearly Borel measurable σ-field. A nearly Borel measurable function u on M is said to be subharmonic resp. harmonic, superharmonic) in D with respect 3

4 to the process X) if for any relatively compact subset U D, t ux t τu ) is a uniformly integrable submartingale resp. martingale, supermartingale) under P x for E-q.e. x U. Here E-q.e. stands for E-quasi-everywhere, meaning it holds outside a set having zero -capacity with respect to the Dirichlet form E, F); see [CF, FOT] for its definition. For a Borel measurable function u on M, we define its non-local tail Tail φ u; x 0, r) in the ball Bx 0, r) by Tail φ u; x 0, r) := Bx 0,r) c uz) V x 0, dx 0, z))φx 0, dx 0, z)) µdz). We need the following definitions for various forms of elliptic Harnack inequalities. Definition.2. i) We say that elliptic Harnack inequality EHI) holds for the process X, if there exist constants δ 0, ) and c such that for every x 0 M, r > 0 and for every non-negative measurable function u on M that is harmonic in Bx 0, r), ess sup Bx0,δr)u c ess inf Bx0,δr)u. ii) We say that non-local elliptic Harnack inequality EHIφ) holds if there exist constants δ 0, ) and c such that for every x 0 M, R > 0, 0 < r δr, and any measurable function u on M that is non-negative and harmonic in Bx 0, R), ) ess sup Bx0,r)u c ess inf Bx0,r)u + φx 0, r)tail φ u ; x 0, R). iii) We say that non-local weak elliptic Harnack inequality WEHIφ) holds if there exist constants ε, δ 0, ) and c such that for every x 0 M, R > 0, 0 < r δr, and any measurable function u on M that is non-negative and harmonic in Bx 0, R), µbx 0, r)) Bx 0,r) /ε ) u dµ) ε c ess inf Bx0,r)u + φx 0, r)tail φ u ; x 0, R). iv) We say that non-local weak elliptic Harnack inequality WEHI + φ) holds if iii) holds for any measurable function u on M that is non-negative and superharmonic in Bx 0, R). Clearly, EHIφ) = EHI + WEHIφ), and WEHI + φ) = WEHIφ). We note that unlike the diffusion case, one needs to assume in the definition of EHI that the harmonic function u is non-negative on the whole space M because the process X can jump all over the places, as mentioned at the beginning of this section. Remark.3. i) For strongly local Dirichlet forms, EHIφ) is just EHI, and WEHI + φ) resp. WEHIφ)) is simply reduced into the following: there exist constants ε, δ 0, ) and c such that for every x 0 M, 0 < r δr, and for every measurable function u that is non-negative and superharmonic resp. harmonic) in Bx 0, R), /ε u dµ) ε c ess inf µbx 0, r)) Bx0,r)u. Bx 0,r) The above inequality is called weak Harnack inequality for differential operators. This is why WEHIφ) is called weak Harnack inequality in [CKP, CKP2, K]. However for non-local operators this terminology is a bit misleading as it is not implied by EHI. 4

5 ii) Non-local weak) elliptic Harnack inequalities have a term involving the non-local tail of harmonic functions, which are essentially due to the jumps of the symmetric Markov processes. This new formulation of Harnack inequalities without requiring the additional positivity on the whole space but adding a non-local tail term first appeared in [K]. The notion of non-local tail of measurable function is formally introduced in [CKP, CKP2], where non-local weak) elliptic Harnack inequalities and local behaviors of fractional p- Laplacians are investigated. See [DK] and references therein for the background of EHI and WEHI. To state relations among various notions of elliptic Harnack inequalities and their characterizations, we need a few definitions. Definition.4. i) We say J φ holds if there exists a non-negative symmetric function Jx, y) so that for µ µ-almost all x, y M, and Jdx, dy) = Jx, y) µdx) µdy),.7) c V x, dx, y))φx, dx, y)) Jx, y) c 2 V x, dx, y))φx, dx, y))..8) We say that J φ, resp. J φ, ) holds if.7) holds and the upper bound resp. lower bound) in.8) holds for Jx, y). ii) We say that IJ φ, holds if for µ-almost all x M and any r > 0, Jx, Bx, r) c ) c 3 φx, r). Remark.5. Under VD and.5), J φ, implies IJ φ,, see, e.g., [CKW, Lemma 2.] or [CK2, Lemma 2.] for a proof. For the non-local Dirichlet form E, F), we define the carré du-champ operator Γf, g) for f, g F by Γf, g)dx) = fx) fy))gx) gy)) Jdx, dy). y M Clearly Ef, g) = Γf, g)m). For any f F b := F L M, µ), Γf, f) is the unique Borel measure called the energy measure) on M satisfying g dγf, f) = Ef, fg) 2 Ef 2, g), f, g F b. M Let U V be open sets in M with U U V. We say a non-negative bounded measurable function ϕ is a cutoff function for U V, if ϕ = on U, ϕ = 0 on V c and 0 ϕ on M. Definition.6. We say that cutoff Sobolev inequality CSJφ) holds if there exist constants C 0 0, ] and C, C 2 > 0 such that for every 0 < r R, almost all x 0 M and any f F, there exists a cutoff function ϕ F b for Bx 0, R) Bx 0, R + r) so that f 2 dγϕ, ϕ) C fx) fy)) U U 2 Jdx, dy) Bx 0,R++C 0 )r) + C 2 f 2 dµ, φx 0, r) Bx 0,R++C 0 )r) where U = Bx 0, R + r) \ Bx 0, R) and U = Bx 0, R + + C 0 )r) \ Bx 0, R C 0 r). 5

6 CSJφ) is introduced in [CKW], and is used to control the energy of cutoff functions and to characterize the stability of heat kernel estimates for non-local Dirichlet forms. See [CKW, Remark.6] for background on CSJφ). Definition.7. We say that Poincaré inequality PIφ) holds if there exist constants C > 0 and κ such that for any ball B r = Bx 0, r) with x 0 M and r > 0, and for any f F b, f f Br ) 2 dµ Cφx 0, r) fy) fx)) 2 Jdx, dy), B r B κr B κr where f Br = µb r) B r f dµ is the average value of f on B r. We next introduce the modified Faber-Krahn inequality. For any open set D M, let F D be the E -closure in F of F C c D), where E u, u) := Eu, u) + M u2 dµ. Define λ D) = inf {Ef, f) : f F D with f 2 = }, the bottom of the Dirichlet spectrum of L on D. Definition.8. We say that Faber-Krahn inequality FKφ) holds, if there exist positive constants C and ν such that for any ball Bx, r) and any open set D Bx, r), λ D) C φx, r) V x, r)/µd))ν. For a set A M, define the exit time τ A = inf{t > 0 : X t A c }. Definition.9. We say that E φ holds if there is a constant c > such that for all r > 0 and all x M 0, c φx, r) Ex [τ Bx,r) ] c φx, r). We say that E φ, resp. E φ, ) holds if the upper bound resp. lower bound) in the inequality above holds. Definition.0. We say elliptic Hölder regularity EHR) holds for the process X, if there exist constants c > 0, θ 0, ] and ε 0, ) such that for every x 0 M, r > 0 and for every bounded measurable function u on M that is harmonic in Bx 0, r), there is a properly exceptional set N u N so that ) dx, y) θ ux) uy) c u r for any x, y Bx 0, εr) \ N u. Here is the main result of this paper. Theorem.. Assume that the metric measure space M, d, µ) satisfies VD, and φ satisfies.5) and.6). Then we have i) WEHIφ) = EHR; WEHIφ) + J φ + FKφ) + CSJφ) = EHIφ). ii) J φ, + FKφ) + PIφ) + CSJφ) = WEHI + φ). 6

7 iii) EHI + E φ, + J φ, = EHIφ) + FKφ); EHI + E φ + J φ, = PIφ). As a direct consequence of Theorem., we have the following statement. Corollary.2. Assume that the metric measure space M, d, µ) satisfies VD, and φ satisfies.5) and.6). If J φ and E φ hold, then FKφ) + PIφ) + CSJφ) WEHI + φ) WEHIφ) EHIφ) EHI. Proof. It follows from Theorem.ii) that, if J φ, holds, then FKφ) + PIφ) + CSJφ) = WEHI + φ) = WEHIφ). By Proposition 4.5i) below, EHR + E φ, = FKφ). On the other hand, according to Proposition 4.7 below, we have E φ + J φ, = CSJφ). Combining those with Theorem.i), we can obtain that under J φ and E φ, WEHIφ) = EHIφ) = EHI. Furthermore, by Theorem.iii), if J φ, and E φ are satisfied, then EHI = FKφ) + PIφ). As mentioned above, Proposition 4.7 below shows that E φ + J φ, = CSJφ). Thus, if J φ, and E φ are satisfied, then EHI = FKφ) + PIφ) + CSJφ). The proof is complete..2 Stability of elliptic Harnack inequalities In this subsection, we study the stability of EHI under some additional assumptions. We mainly follow the framework of [B]. For open subsets A and B of M with A B that is, A A B), define the relative capacity CapA, B) = inf {Eu, u) : u F, u = E-q.e. on A and u = 0 E-q.e. on B c }. For each x M and r > 0, define Our main assumptions are as follows. Extx, r) = V x, r)/capbx, r), Bx, 2r)). Assumption.3. i) M, µ) satisfies VD and RVD. ii) There is a constant c > 0 such that for all x, y M with dx, y) r, CapBy, r), By, 2r)) c CapBx, r), Bx, 2r)). iii) For any a 0, ], there exists a constant c 2 := c 2,a > 0 such that for all x M and r > 0, CapBx, r), Bx, 2r)) c 2 CapBx, ar), Bx, 2r)). 7

8 iv) There exist constants c 3, c 4 > 0 and β 2 β > 0 such that for all x M and 0 < r R, R ) β Extx, R) R ) β2. c 3 c 4.9) r Extx, r) r Assumption.4. For any bounded, non-empty open set D M, there exist a properly exceptional set N D N and a non-negative measurable function G D x, y) defined for x, y D \ N D such that i) G D x, y) = G D y, x) for all x, y) D \ N D ) D \ N D ) \ diag. ii) for every fixed y D \ N D, the function x G D x, y) is harmonic in D \ N D ) \ {y}. iii) for every measurable f 0 on D, [ τd ] E x fx t ) dt = 0 D G D x, y)fy) µdy), x D \ N D. The function G D x, y) satisfying i) iii) of Assumption.4 is called the Green function of X in D. Remark.5. i) We will see from Lemmas 5.2 and 5.3 below that, under suitable conditions, the quantity Extx, r) defined above is related to the mean exit time from the ball Bx, r) by the process X. Hence, under the conditions, Extx, r) plays the same role of the scaling function φx, r) in the previous subsection. ii) From VD and Assumption.3 ii), iii) and iv), we can deduce that for every a 0, ] and L > 0, there exists a constant c 5 := c a,l,5 such that the following holds for all x, y M with dx, y) r, c 5 CapBy, alr), By, 2Lr)) CapBx, r), Bx, 2r)) c 5CapBy, alr), By, 2Lr))..0) iii) Assumption.3 is the same as [BM, Assumption.6] except that in their paper the corresponding conditions are assumed to hold for r 0, R 0 ] and for 0 < r R R 0 with some R 0 > 0. These conditions are called bounded geometry condition in [BM]. However the setting of [BM] is for strongly local Dirichlet forms with underlying state space M being geodesic. Under these settings and the bounded geometry condition, it is shown in [BM] that there exists an equivalent doubling measure µ on M so that Assumption.3 holds i.e., the bounded geometry condition holds globally in large scale as well). Since harmonicity is invariant under time-changes by strictly increasing continuous additive functionals, this enables them to substantially extend the stability result of elliptic Harnack inequality of Bass [B] for diffusions, which was essentially established under the global bounded geometry condition. However the continuity of the processes i.e. diffusions) and the geodesic property of the underlying state space played a crucial role in [BM]. It is unclear at this stage whether Assumption.3 can be replaced by a bounded geometry condition for non-local Dirichlet forms on general metric measure spaces. The following result gives a stable characterization of EHI. Theorem.6. Under Assumptions.3 and.4, if J Ext holds, then FKExt) + PIExt) + CSJExt) WEHI + Ext) WEHIExt) EHIExt) EHI, where J Ext is J φ with Extx, r) replacing φx, r), and same for other notions. 8

9 .3 Parabolic Harnack inequalities As consequences of the main result of this paper, Theorem. and the stability result of parabolic Harnack inequality in [CKW2, Theorem.7], we will present in this subsection new equivalent characterizations of parabolic Hanack inequality in terms of elliptic Harnack inequalities. In this subsection, we always assume that, for each x M there is a kernel Jx, dy) so that Jdx, dy) = Jx, dy) µdy). We aim to present some equivalent conditions for parabolic Harnack inequalities in terms of elliptic Harnack inequalities, which can be viewed as a complement to [CKW2]. We restrict ourselves to the case that the scale) function φ is independent of x, i.e. in this subsection, φ : R + R + is a strictly increasing continuous function with φ0) = 0, φ) = such that there exist constants c 3, c 4 > 0 and β 2 β > 0 so that c 3 R r ) β φr) φr) c 4 R r ) β2 for all 0 < r R..) We first give the probabilistic definition of parabolic functions in the general context of metric measure spaces. Let Z := {V s, X s } s 0 be the space-time process corresponding to X where V s = V 0 s for all s 0. The filtration generated by Z satisfying the usual conditions will be denoted by { F s ; s 0}. The law of the space-time process s Z s starting from t, x) will be denoted by P t,x). For every open subset D of [0, ) M, define τ D = inf{s > 0 : Z s / D}. We say that a nearly Borel measurable function ut, x) on [0, ) M is parabolic or caloric) in D = a, b) Bx 0, r) for the process X if there is a properly exceptional set N u of the process X so that for every relatively compact open subset U of D, ut, x) = E t,x) uz τu ) for every t, x) U [0, ) M\N u )). We next give definitions of parabolic Harnack inequality and Hölder regularity for parabolic functions. Definition.7. i) We say that parabolic Harnack inequality PHIφ) holds for the process X, if there exist constants 0 < C < C 2 < C 3 < C 4, C 5 > and C 6 > 0 such that for every x 0 M, t 0 0, R > 0 and for every non-negative function u = ut, x) on [0, ) M that is parabolic in cylinder Qt 0, x 0, C 4 φr), C 5 R) := t 0, t 0 + C 4 φr)) Bx 0, C 5 R), ess sup Q u C 6 ess inf Q+ u, where Q := t 0 +C φr), t 0 +C 2 φr)) Bx 0, R) and Q + := t 0 +C 3 φr), t 0 +C 4 φr)) Bx 0, R). ii) We say parabolic Hölder regularity PHRφ) holds for the process X, if there exist constants c > 0, θ 0, ] and ε 0, ) such that for every x 0 M, t 0 0, r > 0 and for every bounded measurable function u = ut, x) that is parabolic in Qt 0, x 0, φr), r), there is a properly exceptional set N u N so that φ ) θ s t ) + dx, y) us, x) ut, y) c ess sup r [t0,t0+φr)] M u for every s, t t 0, t 0 + φεr)) and x, y Bx 0, εr) \ N u. 9

10 Definition.8. We say that UJS holds if there is a symmetric function Jx, y) so that Jx, dy) = Jx, y) µdy), and there is a constant c > 0 such that for µ-a.e. x, y M with x y, c Jx, y) Jz, y) µdz) for every 0 < r dx, y). V x, r) Bx,r) 2 We define EHR, E φ, E φ,, J φ,, PIφ) and CSJφ) similarly as in previous subsections but with φr) in place of φx, r). The following stability result of PHIφ) is recently established in [CKW2]. Theorem.9. [CKW2, Theorem.7]) Suppose that the metric measure space M, d, µ) satisfies VD and RVD, and φ satisfies.). Then the following are equivalent: i) PHIφ). ii) PHRφ) + E φ, + UJS. iii) EHR + E φ + UJS. iv) J φ, + PIφ) + CSJφ) + UJS. As a consequence of Theorems. and.9, we have the following statement for the equivalence of PHIφ) in terms of EHI. Theorem.20. Suppose that the metric measure space M, d, µ) satisfies VD and RVD, and φ satisfies.). Then the following are equivalent: i) PHIφ). ii) WEHI + φ) + E φ + UJS. iii) WEHIφ) + E φ + UJS. iv) EHIφ) + E φ + UJS. v) EHI + E φ + UJS + J φ,. Proof. As indicated in Theorem.9, under VD, RVD and.), PHIφ) J φ, + PIφ) + CSJφ) + UJS = E φ. Then, by Theorem.ii), i) = ii). ii) = iii) is clear. iii) = i) follows from Theorem.i) and Theorem.9iii). Obviously, i) = v) is a consequence of Theorem.9 i), iii) and iv). v) = iv) follows from Theorem.iii). iv) = iii) is trivial. This completes the proof. The remainder of this paper is mainly concerned with the proof of Theorem., the main result of this paper. It is organized as follows. The proofs of Theorem.i), ii) and iii) are given in the next three sections, respectively. In Section 5, we study the relations between the mean of exit time and relative capacity. In particular, the proof of Theorem.6 is given there. Finally, a class of symmetric jump processes of variable orders on R d with state-dependent scaling functions are given in Section 6, for which we apply the main results of this paper to show that all the elliptic Harnack inequalities hold for these processes. In this paper, we use := as a way of definition. For two functions f and g, notation f g means that there is a constant c so that g/c f cg. 0

11 2 Elliptic Harnack inequalities and Hölder regularity In this section, we assume that µ and φ satisfy VD,.5) and.6), respectively. We will prove that WEHIφ) implies a priori Hölder regularity for harmonic functions, and study the relation between WEHIφ) and EHIφ). 2. WEHIφ) = EHR In this part, we will show that the weak elliptic Harnack inequality implies regularity estimates of harmonic functions in Hölder spaces. We mainly follow the strategy of [DK, Theorem.4], part of which is originally due to [M, Sil]. Theorem 2.. Suppose that VD,.5) and WEHIφ) hold. Then there exist constants β 0, ) and c > 0 such that for any x 0 M, r > 0 and harmonic function u on Bx 0, r), In particular, EHR holds. ρ ) β ess osc Bx0,ρ)u c u, 0 < ρ r. 2.) r Proof. ) Without loss of generality, we assume the harmonic function u is bounded. Throughout the proof, we fix x 0 M, and denote by B r = Bx 0, r) for any r > 0. For a given bounded harmonic function u on B r, we will construct an increasing sequence m n ) n of positive numbers and a decreasing sequence M n ) n that satisfy for any n N {0}, m n ux) M n M n m n = Kθ nβ. for x B rθ n; 2.2) Here K = M 0 m 0 [0, 2 u ] with M 0 = u and m 0 = ess inf M u, and the constants θ = θδ) δ and β = βδ) 0, ) are determined later so that 2 λ θ β for λ := 2 +/ε c) 0, ), 2.3) 2 where ε, δ 0, ) and c are the constants in the definition of WEHIφ). Let us first show that how this construction proves the first desired assertion 2.). Given ρ < r, there is a j N {0} such that From 2.2), we conclude rθ j ρ < rθ j. ess osc Bρ u ess osc Brθ j u M j m j = Kθ jβ 2θ β u ρ r ) β. Set M n = M 0 and m n = m 0 for any n N. Assume that there is a k N and there are M n and m n such that 2.2) holds for n k. We need to choose m k, M k such that 2.2) still holds for n = k. Then the desired assertion follows by induction. For any x M, set vx) = ux) M ) k + m k 2θ k )β 2 K.

12 Then the definition of v implies that vx) for almost all x B rθ k ). Given y M with dy, x 0 ) rθ k ), there is a j N such that rθ k+j dy, x 0 ) < rθ k+j+. For such y M and j N, on the one hand, we conclude that K 2θ k )β vy) =uy) M k + m k 2 M k j m k j + m k j M k + m k 2 M k j m k j M k m k 2 Kθ k j )β K 2 θ k )β, where in the equalities above we used the fact that if j > k, then uy) M 0, m k j m 0 and M 0 m 0 Kθ k j )β. That is, On the other hand, similarly, we have i.e. ) dy, vy) 2θ jβ x0 ) β ) K 2θ k )β vy) =uy) M k + m k 2 rθ k m k j M k j + M k j M k + m k 2 M k j m k j ) + M k m k 2 Kθ k j )β + K 2 θ k )β, ) dy, vy) 2θ jβ x0 ) β 2. Now, there are two cases: i) µ {x B rθ k : vx) 0}) µb rθ k)/2. ii) µ {x B rθ k : vx) > 0}) µb rθ k)/2. In case i) we aim to show vz) λ for almost every z B rθ k. If this holds true, then for any z B rθ k, rθ k λ)k uz) θ k )β + M k + m k 2 2 λ)k = θ k )β + M k m k + m k 2 2 λ)k = θ k )β + K 2 2 θ k )β + m k Kθ kβ + m k, 2

13 where the last inequality follows from the first inequality in 2.3). Thus, we set m k = m k and M k = m k + Kθ kβ, and obtain that m k uz) M k for almost every z B rθ k. Consider w = v and note that w 0 in B rθ k ). Since in the present setting there is no killing inside M 0 for the process X, constant functions are harmonic, and so w is also harmonic function. Applying WEHIφ) with w on B rθ k ), we find that µb rθ k) B rθ k w ε du ) /ε ) c ess inf w + φx Brθ k 0, rθ k )Tail φ w ; x 0, rθ k ) ). Note that, since the constant c in the definition of WEHIφ) may depend on δ and ε, in the above inequality the constant c = c could also depend on δ and ε, thanks to the fact that θ δ. Under case i), µb rθ k) B rθ k On the other hand, by 2.4), Remark.5 and.5), φx 0, rθ k ) )Tail φ w ; x 0, rθ k ) ) φx 0, rθ k ) vz)) ) V x 0, dx 0, z))φx 0, dx 0, z)) µdz) φx 0, rθ k ) ) φx 0, rθ k ) ) 2φx 0, rθ k ) ) j= B c rθ k ) j= j= B rθ k+j+ \B rθ k+j c 2 φx 0, rθ k ) ) j= c 3 θ jβ θ jβ ), j= B rθ k+j+ \B rθ k+j B rθ k+j+ \B rθ k+j rθ k 2.5) w ε du) /ε 2 /ε. 2.6) vz)) V x 0, dx 0, z))φx 0, dx 0, z)) µdz) vz) )) + V x 0, dx 0, z))φx 0, dx 0, z)) µdz) [ dx0 ), z) β ] V x 0, dx 0, z))φx 0, dx 0, z)) µdz) θ j+)β φx 0, rθ k+j ) where c 3 > 0 is a constant independent of k and r but depend on θ and β from.5). Hence, by.5), 2.5), 2.6) and 2.7), we obtain ess inf Brθ k w c 2 /ε ) c 4 φx 0, rθ k ) )Tail φ w ; x 0, rθ k ) ) c2 /ε ) c 5 θ jβ θ jβ ). Note that all the constants c i i =,..., 5) may depend on θ. Since for any β 0, β ), θ jβ θ jβ ) <, j= j= 3 2.7)

14 we can choose l large enough which is independent of β, θ and only depends on δ) such that for any β 0, β /2), j=l+ θ jβ θ jβ ) j=l+ θ jβ θ jβ /2 ) j=l+ Given l, one can further take β 0, β /2) small enough such that l θ jβ θ jβ ) βlog θ) j= δ jβ /2 < 4c 5 c2 /ε ). l θ jβ β) j βlθ β/2 log θ) < 4c 5 c2 /ε ). j= Without loss of generality we may and do assume that δ in the definition of WEHIφ) is small enough. Thus, the constant β here is also independent of θ and only depends on δ.) Therefore, ess inf Brθ k w 2c2/ε ) = λ. That is, v λ on B rθ k. In case ii), our aim is to show v + λ. This time we set w = + v. Following the arguments above, one sets M k = M k and m k = M k Kθ kβ leading to the desired result. 2) Let δ 0 0, /3). Then for almost all x, y Bx 0, δr), the function u is harmonic on Bx, δ 0 )r). Note that dx, y) 2δ 0 r δ 0 )r. Applying 2.), we have ) dx, y) β ux) uy) ess osc Bx,dx,y)) u c u. δ 0 )r This establishes EHR. Remark 2.2. The argument above in fact shows that WEHIφ) = EHR holds for any general jump processes possibly non-symmetric) that admits no killings inside M. 2.2 WEHIφ) + J φ + FKφ) + CSJφ) = EHIφ) Let D be an open subset of M. Recall that a function f is said to be locally in F D, denoted as f FD loc, if for every relatively compact subset U of D, there is a function g F D such that f = g m-a.e. on U. The following is established in [C]. Lemma 2.3. [C, Lemma 2.6]) Let D be an open subset of M. Suppose u is a function in F loc D that is locally bounded on D and satisfies that U V c uy) Jdx, dy) < 2.8) for any relatively compact open sets U and V of M with Ū V V D. Then for every v F C c D), the expression ux) uy))vx) vy)) Jdx, dy) is well defined and finite; it will still be denoted as Eu, v). 4

15 As noted in [C, 2.3)], since E, F) is a regular Dirichlet form on L 2 M; µ), for any relatively compact open sets U and V with Ū V, there is a function ψ F C cm) such that ψ = on U and ψ = 0 on V c. Consequently, Jdx, dy) = ψx) ψy)) 2 Jdx, dy) Eψ, ψ) <, U V c U V c so each bounded function u satisfies 2.8). We say that a nearly Borel measurable function u on M is E-subharmonic resp. E-harmonic, E-superharmonic) in D if u FD loc that is locally bounded on D, satisfies 2.8) for any relatively compact open sets U and V of M with Ū V V D, and that Eu, ϕ) 0 resp. = 0, 0) for any 0 ϕ F C c D). The following is established in [C, Theorem 2. and Lemma 2.3] first for harmonic functions, and then extended in [ChK, Theorem 2.9] to subharmonic functions. Theorem 2.4. Let D be an open subset of M, and u be a bounded function. Then u is E- harmonic resp. E-subharmonic) in D if and only if u is harmonic resp. subharmonic) in D. The next lemma can be proved by the same argument as that for [CKW, Proposition 2.3]. Lemma 2.5. Assume that VD,.5),.6), J φ, and CSJφ) hold. Then there is a constant c 0 > 0 such that for every 0 < r R and almost all x M, V x, R + r) CapBx, R), Bx, R + r)) c 0. φx, r) Using this lemma, we can establish the following. Lemma 2.6. Let B r = Bx 0, r) for some x 0 M and r > 0. Assume that u is a bounded and E-superharmonic function on B R such that u 0 on B R. If VD,.5),.6), J φ, FKφ) and CSJφ) hold, then for any 0 < r < R, φx 0, r)tail φ u + ; x 0, r) c ess sup Br u + φx 0, r)tail φ u ; x 0, R) ), where c > 0 is a constant independent of u, x 0, r and R. Proof. According to J φ,, CSJφ) and Lemma 2.5, we can choose ϕ F B3r/4 related to CapB r/2, B 3r/4 ) such that Let k = ess sup Br u and w = u 2k. wϕ 2 F B3r/4 with w < 0 on B r, Eϕ, ϕ) 2CapB r/2, B 3r/4 ) c V x 0, r) φx 0, r). 2.9) Since u is an E-superharmonic function on B R, and 0 E u, wϕ 2) = ux) uy))wx)ϕ 2 x) wy)ϕ 2 y)) Jdx, dy) B r B r + 2 ux) uy))wx)ϕ 2 x) Jdx, dy) B r B c r 5

16 = : I + 2I 2. For any x, y B r, ux) uy))wx)ϕ 2 x) wy)ϕ 2 y)) = wx) wy))wx)ϕ 2 x) wy)ϕ 2 y)) = ϕ 2 x)wx) wy)) 2 + wy)ϕ 2 x) ϕ 2 y))wx) wy)) ϕ 2 x)wx) wy)) 2 8 ϕx) + ϕy))2 wx) wy)) 2 2w 2 y)ϕx) ϕy)) 2, where we used the fact that ab 8 a2 + 2b 2) for all a, b R in the inequality above. Hence, I ϕ 2 x)wx) wy)) 2 Jdx, dy) B r B r ϕx) + ϕy)) 2 wx) wy)) 2 Jdx, dy) 8 B r B r 2 w 2 y)ϕx) ϕy)) 2 Jdx, dy) B r B r ϕ 2 x)wx) wy)) 2 Jdx, dy) 2 B r B r 8k 2 B r B r ϕx) ϕy)) 2 Jdx, dy) 8k 2 B r B r ϕx) ϕy)) 2 Jdx, dy), where in the second inequality we have used the symmetry property of Jdx, dy) and the fact that w 2 4k 2 on B r. On the other hand, by the definition of w, it is easy to see that for any x B r and y / B r ux) uy))wx) kuy) k) + 2k {uy) k} ux) uy)) + kuy) k) + 2kux) uy)) +, and so I 2 B r B c r B r B c r = : I 2 I 22. kuy) k) + ϕ 2 x) Jdx, dy) 2kux) uy)) + ϕ 2 x) Jdx, dy) Furthermore, since uy) k) + u + y) k, we find that I 2 k u + y)ϕ 2 x) Jdx, dy) k 2 B r B c r kµb r/2 ) inf x B r/2 B c r B r B c r u + y) Jx, dy) k 2 c kv x 0, r)tail φ u + ; x 0, r) k 2 6 B r B c r ϕ 2 x) Jdx, dy) ϕ 2 x) Jdx, dy) B r B c r ϕ 2 x) Jdx, dy),

17 where in the second inequality we have used the fact that ϕ = on B r/2, and in the last inequality we have used J φ, and the fact that for all x B r/2 and z B c r, V x, dx, z)) φx, dx, z)) V x 0, dx 0, z)) φx 0, dx 0, z)) c + dx, x ) d2 +β 2 0) c, dx 0, z) thanks to VD,.5) and.6). Also, since u 0 on B R, we can check that I 22 2k kϕ 2 x) Jdx, dy) + 2k k + u y))ϕ 2 x) Jdx, dy) 2k 2 B r B R \B r) B r B c r B r B c R ϕ 2 x) Jdx, dy) + c 2 k 2 V x 0, r) φx 0, r) + c 2kV x 0, r)tail φ u ; x 0, R), where the second term of the last inequality follows from Remark.5 and.6), and in the third term we have used J φ,. By the estimates for I 2 and I 22, we get that I 2 3k 2 ϕ 2 x) Jdx, dy) + c kv x 0, r)tail φ u + ; x 0, r) B r B c r c 2 k 2 V x 0, r) φx 0, r) c 2kV x 0, r)tail φ u ; x 0, R). This along with the estimate for I yields that V x0, r) V x 0, r)tail φ u + ; x 0, r) c 3 [k φx 0, r) ) ] + Eϕ, ϕ) + V x 0, r)tail φ u ; x 0, R). Then, combining this inequality with 2.9) proves the desired assertion. We also need the following result. Since the proof is essentially the same as that of [CKW, Proposition 4.0], we omit it here. Proposition 2.7. Let x 0 M and R > 0. Assume VD,.5),.6), J φ,, FKφ) and CSJφ) hold, and let u be a bounded E-subharmonic in Bx 0, R). Then for any δ > 0, ess sup Bx0,R/2)u c + δ ) /ν /2 u dµ) 2 + δφx 0, R)Tail φ u; x 0, R/2), V x 0, R) Bx 0,R) where ν is the constant in FKφ), and c > 0 is a constant independent of x 0, R, δ and u. We are in a position to present the main statement in this subsection. Theorem 2.8. Let B r x 0 ) = Bx 0, r) for some x 0 M and r > 0. Assume that u is a bounded and E-harmonic function on B R x 0 ) such that u 0 on B R x 0 ). Assume that VD,.5),.6), J φ, FKφ) and CSJφ), and WEHIφ) hold. Then the following estimate holds for any 0 < r < δ 0 R, ess sup Br/2 x 0 )u c ess inf Brx 0 )u + φx 0, r)tail φ u ; x 0, R) ), where δ 0 0, ) is the constant δ in WEHIφ) and c > 0 is a constant independent of x 0, r, R and u. This is, WEHIφ) + J φ + FKφ) + CSJφ) = EHIφ). 7

18 Proof. Note that u + is a bounded and E-subharmonic function on B R x 0 ). According to Proposition 2.7, for any 0 < δ < and 0 < ρ < R, ) /2 ess sup Bρ/2 x 0 )u c δφx 0, ρ)tail φ u + ; x 0, ρ/2) + δ /2ν) u 2 + dµ, V x 0, ρ) B ρx 0 ) where c > 0 is a constant independent of x 0, ρ, u and δ. The inequality above along with Lemma 2.6 yields that ) /2 ess sup Bρ/2 x 0 )u c 2 δ /2ν) u 2 + dµ V x 0, ρ) B ρx 0 ) ) + δess sup Bρx0 )u + δφx 0, ρ)tail φ u ; x 0, R). For any /2 σ σ and z B σ rx 0 ), applying the inequality above with B ρ x 0 ) = B σ σ )rz), we get that there is a constant c 3 > such that δ /2ν) ) /2 uz) c 3 σ σ ) d u 2 dµ 2/2 V x 0, σr) B σrx 0 ) ) + δess sup Bσrx0 )u + δφx 0, r)tail φ u ; x 0, R), where we have used the facts that B σ σ )rz) B σr x 0 ) for any z B σ rx 0 ), and V x 0, σr) V z, σ σ )r) c + dx ) 0, z) + σr d2 σ σ c + σr + ) σ r d2 )r σ σ )r thanks to VD. Therefore, c σ σ ) d 2, δ /2ν) ) /2 ess sup Bσ r x 0 )u c 3 σ σ ) d u 2 dµ 2/2 V x 0, σr) B σrx 0 ) ) + δess sup Bσrx0 )u + δφx 0, r)tail φ u ; x 0, R). In particular, choosing δ = 4c 3 ess sup Bσ r x 0 )u 4 ess sup B σrx 0 )u c 4 + σ σ ) d 2/2 V x 0, σr) in the inequality above, we arrive at B σrx 0 ) u 2 dµ) /2 + c 4 φx 0, r)tail φ u ; x 0, R). Since c 4 σ σ ) d 2/2 V x 0, σr) B σrx 0 ) u 2 dµ ) /2 8

19 c 4 ess sup Bσrx0 )u) 2 q)/2 σ σ ) d 2/2 V x 0, σr) B /2 σrx 0 ) 4 ess sup c 4 B σrx 0 )u + σ σ ) d 2/q V x 0, σr) ) /2 u q dµ B σrx 0 ) u q dµ) /q, where in the last inequality we applied the standard Young inequality with exponent 2/q and 2/2 q) with any 0 < q < 2, we have for any 0 < q < 2 and /2 σ σ, ess sup Bσ r x 0 )u 2 ess sup B σrx 0 )u + [ /q ] c 5 σ σ ) d u dµ) q + φx 0, r)tail φ u ; x 0, R) 2/q V x 0, σr) B σrx 0 ) 2 ess sup B σrx 0 )u + c [ /q ] 5 σ σ ) d u dµ) q + φx 0, r)tail φ u ; x 0, R). 2/q V x 0, r) B rx 0 ) According to Lemma 2.9 below, we find that [ /q ] ess sup Br/2 x 0 )u c 6 u dµ) q + φx 0, r)tail φ u ; x 0, R). V x 0, r) B rx 0 ) To conclude the proof, we combine the above inequality with WEHIφ) and Theorem 2.4, by setting q = ε. The following lemma is taken from [GG, Lemma.], which has been used in the proof above. Lemma 2.9. Let ft) be a non-negative bounded function defined for 0 T 0 t T. Suppose that for T 0 t s T we have ft) As t) α + B + θfs), where A, B, α, θ are non-negative constants, and θ <. Then there exists a positive constant c depending only on α and θ such that for every T 0 r R T, we have ) fr) c AR r) α + B. 3 Sufficient condition for WEHI + φ) In this section, we will establish the following, which gives a sufficient condition for WEHI + φ). Theorem 3.. Assume that VD,.5),.6), J φ,, FKφ), PIφ) and CSJφ) hold. Then, WEHI + φ) holds. More precisely, there exist constants ε 0, ) and c such that for all x 0 M, 0 < r < R/60κ) and any bounded E-superharmonic function u on B R := Bx 0, R) with u 0 on B R, ) /ε ) u ε dµ c ess inf Br u + φx 0, r)tail φ u ; x 0, R), µb r ) B r where κ is the constant in PIφ) and B r = Bx 0, r). 9

20 Throughout this section, we always assume that µ and φ satisfy VD,.5) and.6), respectively. To prove Theorem 3. we mainly follow [CKP2], which is originally due to [DT]. Since we essentially make use of CSJφ), some nontrivial modifications are required. We begin with the following result, which easily follows from [CKW2, Corollary 4.2]. Lemma 3.2. Let B r = Bx 0, r) for some x 0 M and r > 0. Assume that u is a bounded and E-superharmonic function on B R such that u 0 on B R. For any a, l > 0 and b >, define v = [ a + l )] log log b. u + l + If VD,.5),.6), J φ,, PIφ) and CSJφ) hold, then for any l > 0 and 0 < r R/2κ), v v Br ) 2 dµ c + φx ) 0, r)tail φ u ; x 0, R), V x 0, r) B r l where κ is the constant in PIφ), v Br = µb r) B r v dµ and c is a constant independent of u, x 0, r, R and l. Lemma 3.3. Let B r = Bx 0, r) for some x 0 M and r > 0. Assume that u is a bounded and E-superharmonic function on B R such that u 0 on B R. Assume that VD,.5),.6), J φ,, FKφ), PIφ) and CSJφ) hold. Suppose that there exist constants λ > 0 and σ 0, ] such that µb r {u λ}) σµb r ) 3.) for some r with 0 < r < R/2κ), where κ is the constant in PIφ). Then there exists a constant c > 0 such that µb 6r {u 2δλ 2 φx 0, r)tail φ u ; x 0, R)}) µb 6r ) c σ log 2δ holds for all δ 0, /4), where c is a constant independent of u, x 0, r, R, σ, λ and δ. Proof. Taking l = 2 φx 0, r)tail φ u ; x 0, R), a = λ with λ > 0, and b = 2δ with δ 0, /4) in Lemma 3.2, we get that for all λ > 0 and 0 < r < R/2κ), where ) /2 v v B6r dµ v v B6r ) 2 dµ c, 3.2) V x 0, 6r) B 6r V x 0, 6r) B 6r {[ λ + l ) ] v = min log, log }. u + l + 2δ Notice that by the definition of v, we have {v = 0} = {u λ}. Hence, by 3.) and VD, for some r with 0 < r < R/2κ), µb 6r {v = 0}) c σµb 6r ) and so log 2δ = log 2δ ) µb 6r {v = 0}) v dµ B 6r {v=0} 20

21 c σ c σ µb 6r ) B 6r log 2δ ) v B6r. log 2δ ) v dµ Thus, integrating the previous inequality over B 6r {v = log 2δ }, we obtain log ) { µ B 6r v = log } ) 2δ 2δ c σ c σ B 6r {v=log 2δ } B 6r v v B6r dµ c 2 σ V x 0, 6r), log ) 2σ v B 6r dµ where in the last inequality we have used 3.2). Therefore, for all 0 < δ < /4, µb 6r {u + l 2δλ + l)}) c 2 σ log V x 0, 6r), 2δ which proves the desired assertion. For any x M and r > 0, set B r x) = Bx, r). For a ball B M and a function w on B, write Iw, B) = w 2 dµ. The following lemma can be proved similarly as the of [CKW, Lemma 4.8]. Lemma 3.4. Suppose VD,.5),.6), J φ,, FKφ) and CSJφ) hold. For x 0 M, R, r, r 2 > 0 with r [ 2 R, R] and r + r 2 R, let u be an E-subharmonic function on B R x 0 ), and v = u θ) + for some θ > 0. Set I 0 = Iu, B r +r 2 x 0 )) and I = Iv, B r x 0 )). We have c I θ 2ν V x 0, R) ν I+ν 0 + r ) [ β2 + + r ) ] d2 +β 2 β φx 0, R)Tail φ u; x 0, R/2), r 2 r 2 θ where ν is the constant in FKφ), d 2 is the constant in.4), β and β 2 are the constants in.5), and c is a constant independent of θ, x 0, R, r, r 2 and u. We also need the following elementary iteration lemma, see, e.g., [Giu, Lemma 7.] or [CKW, Lemma 4.9]. Lemma 3.5. Let β > 0 and let {A j } be a sequence of real positive numbers such that with c 0 > 0 and b >. If then we have which in particular yields lim j A j = 0. B A j+ c 0 b j A +β j, j A 0 c /β 0 b /β2, A j b j/β A 0, j 0, 2

22 The following proposition gives us the infimum of the superharmonic function. This extends the analogous expansion of positivity in the local setting, which is a key step towards WEHI + φ). Proposition 3.6. Let B r = Bx 0, r) for some x 0 M and any r > 0. Assume that u is a bounded and E-superharmonic function on B R such that u 0 on B R. Assume that VD,.5),.6), J φ,, FKφ), PIφ) and CSJφ) hold. Suppose that there exist constants λ > 0 and σ 0, ] such that µb r {u λ}) σµb r ) 3.3) for some r satisfying 0 < r < R/2κ), where κ is the constant in PIφ). Then, there exists a constant δ 0, /4) depending on σ but independent of λ, r, R, x 0 and u, such that ess inf B4r u δλ φx 0, r)tail φ u ; x 0, R). 3.4) Proof. Without loss of generality, we may and do assume that φx 0, r)tail φ u ; x 0, R) δλ; 3.5) otherwise the conclusion is trivial due to the fact that u 0 on B R. For any j 0, define Then, by 3.5) we see that Lemma 3.3 implies that In the following, let us denote by l j = δλ + 2 j δλ, r j = 4r + 2 j r. l 0 = 3 2 δλ 2δλ 2 φx 0, r)tail φ u ; x 0, R). µb 6r {u l 0 }) µb 6r ) c σ log. 3.6) 2δ B j = B rj, w j = l j u) +, A j = µb j {u l j }). µb j ) Note that, u is an E-subharmonic function on B R. Then, we have by Lemma 3.4 that A j+2 l j+ l j+2 ) 2 = l j+ l j+2 ) 2 dµ µb j+2 ) B j+2 {u l j+2 } wj+ 2 dµ µb j+2 ) B j+2 ) +ν ) c 2 l j l j+ ) 2ν w 2 r β2 j+2 j dµ µb j+ ) B j+ r j+ r j+2 [ ) r d2 +β 2 β j+2 + φx 0, r j+)tail φ w j ; x 0, r j+)] l j l j+ r j+ r j+2 ) c 3 [ δλ) 2 ] +ν β2 [2 j 2 j )δλ] 2ν A j 2 j 2 j 22

23 [ + 2 j 2 j )δλ c 4 δλ) 2 A +ν j 2 +2ν+d 2+2β 2 β )j ) d2 +β 2 β 2 j 2 j φx 0, r j+)tail φ w j ; x 0, r j+)] + ) δλ φx 0, r j+ )Tail φ w j ; x 0, r j+ ), where ν is the constant in FKφ), and in the third inequality we have used the facts that w j l j 3δλ/2 and B j+ w 2 j dµ c δλ) 2 µb j+ {w j 0}) c δλ) 2 µb j {u l j }). Note that φx 0, r j+ )Tail φ w j ; x 0, r j+ ) w j z) = φx 0, r j+ ) Bj+ c V x 0, dx 0, z))φx 0, dx 0, z)) µdz) l j + u z) φx 0, r j+ ) Bj+ c V x 0, dx 0, z))φx 0, dx 0, z)) µdz) [ l j = φx 0, r j+ ) B R \B j+ V x 0, dx 0, z))φx 0, dx 0, z)) µdz) ] l j + u z) + BR c V x 0, dx 0, z))φx 0, dx 0, z)) µdz) c 5 l j + φx ) 0, r j+ ) φx 0, R) l j + φx 0, r)tail φ u ; x 0, R) c 6 δλ, where in the second equality we have used the fact that u 0 on B R, in the second inequality we used Remark.5, and the last inequality follows from 3.5). According to all the estimates above, we see that there is a constant c 7 > 0 such that for all j 0, A j+2 c 7 A +ν j 2 3+2ν+d 2+2β 2 β )j. Let c = c /ν ν+d 2+2β 2 β )/ν 2 and choose the constant δ 0, /4 ) such that c σ log 2δ c, then, by 3.6), A 0 c = c /ν ν+d 2+2β 2 β )/ν 2. According to Lemma 3.5, we can deduce that lim i A i = 0. Therefore, u δλ on B 4r, from which the desired assertion follows easily. Remark 3.7. Proposition 3.6 contains [GHH, Lemma 4.5] as a special case. Among other things, in [GHH, Lemma 4.5], it is assumed that φx, r) = r α for J φ. Inequality 3.4) under condition 3.3) is called a weak Harnack inequality in [GHH]. 23

24 Below is a Krylov-Safonov covering lemma on metric measure spaces, whose proof is essentially taken from [KS, Lemma 7.2]. Note that the difference of the following definition of [E] η from that in [KS, Lemma 7.2] is that here we impose the restriction of 0 < ρ < r and change the constant 3 to 5. For the sake of completeness, we present the proof here. Lemma 3.8. Suppose that VD holds. Let B r x 0 ) = Bx 0, r) for some x 0 M and any r > 0. Let E B r x 0 ) be a measurable set. For any η 0, ), define [E] η = { B 5ρ x) B r x 0 ) : x B r x 0 ) and µe B } 5ρx)) > η. µb ρ x)) 0<ρ<r Then, either or [E] η = B r x 0 ) µ[e] η ) η µe). Proof. Define a maximal operator A : B r x 0 ) [0, ) as follows We claim that Ax) = sup y B rx 0 ),x B 5ρ y),0<ρ<r µe B 5ρ y)). µb ρ y)) [E] η = {x B r x 0 ) : Ax) > η} for any η 0, ). Indeed, let x B r x 0 ) with Ax) > η. Then, there is a ball B ρ y) with 0 < ρ < r, y B r x 0 ) and x B 5ρ y) such that µe B 5ρy)) µb ρy)) > η. This means that { x B 5ρ y) B r x 0 ) : y B r x 0 ) and µe B } 5ρy)) > η [E] η. µb ρ y)) On the other hand, if x [E] η, then there is a ball B ρ y) with 0 < ρ < r, y B r x 0 ) and x B 5ρ y) such that µe B 5ρy)) µb ρy)) > η. This implies that Ax) > η. Suppose that B r x 0 ) \ [E] η. The set [E] η is open by definition. We cover [E] η by ball B rx x), where x [E] η and r x = dx, B r x 0 ) \ [E] η )/2 0, r). By the Vitali covering lemma, there are countably many pairwise disjoint balls B ri x i ), where r i = r xi for all i, such that [E] η i= B 5r i x i ). Note that, B 5ri x i ) B r x 0 ) \ [E] η ) for all i, and so there is a point y i B 5ri x i ) B r x 0 ) \ [E] η ). In particular, Ay i ) η for all i. Since y i B r x 0 ), x i B 5ri y i ) and 0 < r i < r, we conclude that µe B 5ri x i )) ηµb ri x i )). If y is the density point E i.e. y E such that lim ρ 0 µe B ρy)) µb ρy)) = ), then lim inf ρ 0 µe B 5ρ y)) µb ρ y)) µe B ρ y)) lim = > η. ρ 0 µb ρ y)) The VD condition implies a covering theorem, here referred to as the Vitali covering theorem. Indeed, given any collection of balls with uniformly bounded radius, there exists a pairwise disjoint, countable subcollection of balls, whose 5-dilates cover the union of the original collection. See [He, Chapter ] for more details. 24

25 Since µ-almost every point E is a density point; that is, for almost all x E, it holds that µe B lim ρx)) ρ 0 µb ρx)) =, which follows from VD and the Lebesgue differentiation theorem see [He, Theorem.8]), we observe that µ-almost every point of E belongs to [E] η for every η 0, ). From this it follows that µe) = µe [E] η ) µe B 5ri x i )) η µb ri x i )) ηµ[e] η ). i= The above inequality yields the desired result. We now are in a position to present the Proof of Theorem 3.. Fix η 0, ). Let us define for any t > 0 and i 0 A i t = where δ 0, ) is determined later and Obviously we have A i t that B 5ρ x) B r x 0 ) [A i t i= { x B r x 0 ) : ux) > tδ i T }, δ T = φx 0, 5r)Tail φ u ; x 0, R). A i t for all i. If there are a point x B r x 0 ) and 0 < ρ < r such ] η, then, by the definition of [A i t ] η and VD, µa i t B 5ρ x)) ηµb ρ x)) c ηµb 5ρ x)), where c is a positive constant independent of η, x 0 and ρ. Below, let δ be the constant given in Proposition 3.6 corresponding to the factor c η. Applying Proposition 3.6 with λ = tδ i T δ and σ = c η, we get that ess inf B20ρ x)u >δ tδ i T ) φx 0, 5ρ)Tail φ u ; x 0, R) δ δ tδ i T ) T δ =tδ i T δ. Hence, if B 5ρ x) is one of the balls to make up to the set [At i ] η in Lemma 3.8, then B 5ρ x) B r x 0 ) A i t, which implies that [A i t ] η A i t. By Lemma 3.8, we must have either A i t = B r x 0 ) since A i t B r x 0 ) for all i 0) or We choose an integer j so that µa i t) η µai t ). 3.7) η j < µa 0 t )/µb r x 0 )) η j. Suppose first that A j t B r x 0 ). Then, by the fact that At i A i t for all i, we have A k t B r x 0 ) for all 0 k j. Hence, according to 3.7), we obtain that µa j t ) η µaj 2 t ) η j µa0 t ) ηµb r x 0 )). 25

26 Note that, the inequality holds trivially for the case that A j t = B r x 0 ), thanks to the fact that η 0, ). Therefore, according to Proposition 3.6 again, we have ess inf B4r x 0 )u c tδ j T δ ) T c tδ j c 2T δ ) µa 0 /γ c t t ) c 2T µb r x 0 )) δ, where c is the constant given in Proposition 3.6 corresponding to the factor δ, and γ = log δ η. This is, µa 0 t ) µb r x 0 )) c 3 t γ ess inf B4r x 0 )u + T ) γ. δ By Cavalieri s principle, we have for any 0 < ε < γ and a > 0, u ɛ dµ = ε t ε µb rx 0 ) {u > t}) dt µb r x 0 )) µb r x 0 )) B rx 0 ) In particular, taking we finally get that 0 ε t ε µa 0 t ) 0 µb r x 0 )) dt [ a ε t ε dt + c 3 ess inf B4r x 0 )u + T ) γ 0 δ a ) γ ] c 4 [a ε + ess inf B4r x 0 )u + a ε γ. a = ess inf B4r x 0 )u + T δ T δ, u ɛ dµ c 4 ess inf µb r x 0 )) B4r x 0 )u + T ) ε. B rx 0 ) δ This along with.5) concludes the proof. ] t ε γ dt Remark 3.9. WEHI + φ) is equivalent to the inequality 3.4) under VD and condition 3.3). Indeed, the proof above shows that 3.4) under VD and condition 3.3) implies WEHI + φ). Conversely, assume WEHI + φ) holds. Under condition 3.3) we have ) /ε u ε dµ µb r ) B r µb r ) Plugging this into WEHI + φ) yields 3.4). 4 Implications of EHI B r {u λ} u ε dµ) /ε λσ /ε. In this section, we first study the relation between EHI and EHIφ), and then show that under some conditions EHI implies PIφ). 26