Boundary Harna k prin iple and ellipti Harna k inequality 7 Martin T. Barlow∗, Mathav Murugan† 1 0 2 January 10, 2017 n a J 7 Abstract ] R We prove a scale-invariant boundary Harnack principle for inner uniform do- P mains over a large family of Dirichlet spaces. A novel feature of our work is that . h our assumptions are robust to time changes of the corresponding diffusions. In t a particular, we do not assume volume doubling property for the symmetric measure. m [ Keywords: BoundaryHarnack principle,EllipticHarnack inequality, Martinbound- ary. 1 v Subject Classification (2010): 31B25, 31B05 2 8 7 1 Introduction 1 0 . 1 Let (X,d) be a metric space, and assume that associated with this space is a structure 0 7 which gives a family of harmonic functions on domains D ⊂ X. (For example, Rd with 1 the usual definition of harmonic functions.) The elliptic Harnack inequality (EHI) holds : v if there exists a constant C such that whenever h is non-negative and harmonic in a ball H i X B(x,r) then, writing 1B = B(x,r/2), 2 r a esssup h ≤ C essinf h. (1.1) 1 H 1 B B 2 2 Thus the EHI controls harmonic functions in a domain D away from the boundary ∂D. On the other hand, the boundary Harnack principle (BHP) gives control of the ratio of two positive harmonic functions at boundary points of a domain. The BHP given in [Anc] states that if D ⊂ Rd is a Lipschitz domain, ξ ∈ ∂D, r > 0 is small enough, then for any pair u,v of harmonic functions in D which vanish on ∂D ∩B(ξ,2r), u(x) u(y) ≤ C for x,y ∈ D ∩B(ξ,r). (1.2) v(x) v(y) ∗Research partially supported by NSERC (Canada) †Research partially supported by NSERC (Canada) and the Pacific Institute for the Mathematical Sciences 1 The BHP is a key component in understanding the behaviour of harmonic functions near the boundary. It will in general lead to a characterisation of the Martin boundary, and there is a close connection between BHP and a Carleson estimate – see [ALM, Aik08]. (See [Aik08] also for a discussion of different kinds of BHP.) The results in [Anc] have been extended in several ways. The first direction has been to weaken the smoothness hypotheses on the domain D; for example [Aik01] proves a BHP for inner uniform domains in Euclidean space. A second direction is to consider functions which are harmonic with respect to more general operators. The standard Laplacian is the (infinitesimal) generator of the semigroup for Brownian motion, and it is natural to ask about the BHP for more general Markov processes with values in a metric space (X,d). [GyS] prove a BHP for inner uniform domains in a measure metric space (X,d,m) with a Dirichlet form which satisfies the standard parabolic Harnack inequality PHI(2). These results are extended in [L] to spaces which satisfy a parabolic Harnack inequality with anomalous space-time scaling. In most case the BHP has been proved for Markov processes which are symmetric, but see [LS] for the BHPfor some more generalprocesses. Allthepaperscitedabovestudytheharmonicfunctionsassociatedwith continuous Markov processes: see [Bog, BKK] for BHP associated with jump processes. The starting point for this paper is the observation that while the BHP is a purely elliptic result, previous proofs all use parabolic data, or more precisely information on the heat kernel of the process. This use is implicit in [Aik01], which just looks at the standard Laplacian on Rd, but is explicit in [GyS, L], where Green’s functions are controlled, from above and below by expressions of the form Ψ(r)/µ(B(x,r)): here Ψ is the space-time scaling function. The main result of this paper (Theorem 5.1) is that, provided the underlying metric space and process have enough local regularity, then the BHP holds for inner uniform domains whenever the elliptic Harnack inequality holds. Since the EHI is weaker than the PHI, our result extends the BHP to a wider class of spaces; also our approach has the advantage that we can dispense with unnecessary parabolic information. Our main result provides new examples of differential operators that satisfy BHP even in Rn – see [GS, eq. (2.1) and Example 6.14]. The contents of the paper are as follows. In Section 2 we give the definition and basic properties of inner uniform domains in length spaces. Section 3 reviews the properties of Dirichlet forms and the associated Hunt processes. In Section 4 we give the definition of harmonic function in our context, and state the additional regularity properties which we will need. We show that these lead to the existence of Green’s functions, and we prove the essential technical result that Green’s functions are locally in the domain of the Dirichlet form – see Lemma 4.17. Some key comparisons of Green’s functions, which follow from the EHI, and were proved in [BM], are given in Proposition 4.18. After these rather lengthy preliminaries, in Section 5 we state and prove our main result Theorem 5.1. Our argument follows that of Aikawa [Aik01] (see also [GyS, L]), except that at key points in the argument we use Green’s function comparisons from Proposition 4.18 rather than bounds which come from heat kernel estimates. We use c,c′,C,C′ for strictly positive constants, which may change value from line 2 to line. Constants with numerical subscripts will keep the same value in each argument, while those with letter subscripts will be regarded as constant throughout the paper. The notation C = C (a) means that the constant C depends only on the constant a. 0 0 0 2 Inner uniform domains In this section, we introduce the geometric assumptions on the underlying metric space, and the corresponding domains. Definition 2.1 (Length space). Let (X,d) be a metric space. The length L(γ) ∈ [0,∞] of a continuous curve γ : [0,1] → X is given by L(γ) = sup d(γ(t ),γ(t )), i−1 i i X where the supremum is taken over all partitions 0 = t < t < ... < t = 1 of [0,1]. 0 1 k Clearly L(γ) ≥ d(γ(0),γ(1)). A metric space is a length space if d(x,y) is equal to the infimum of the lengths of continuous curves joining x and y. For the rest of this paper, we assume that (X,d) is a complete, separable, locally compact, length space. By the Hopf–Rinow–Cohn–Vossen theorem (cf. [BBI, Theorem 2.5.28]) every closed metric ball in (X,d) is compact. It also follows that there exists a geodesic path γ(x,y) (not necessarily unique) between any two points x,y ∈ X. We write B(x,r) = {y ∈ X : d(x,y) < r} for open balls in (X,d). Next, we introduce the intrinsic distance d induced by an open set U ⊂ X. U Definition 2.2 (Intrinsic distance). Let U ⊂ X be a connected open subset. We define the intrinsic distance d by U d (x,y) = inf{L(γ) : γ : [0,1] → U continuous, γ(0) = x,γ(1) = y}. U It is well-known that (U,d ) is a length metric space (cf. [BBI, Exercise 2.4.15]). We U now consider its completion. Definition 2.3 (Balls in intrinsic metric). Let U ⊂ X be a connected and open subset of the length space (X,d). Let U denote the completion of (U,d ), equipped with the U natural extension of d to U ×U. For x ∈ U we define U e BeUe(x,er) = y ∈ Ue : dU(x,y) < r . n o Set e BU(x,r) = U ∩BUe(x,r). If x ∈ U, then B (x,r) simply corresponds to the open ball in (U,d ). However, the U U definition of B (x,r) also makes sense for x ∈ U \U. U 3 e Definition 2.4 (Boundary and distance to the boundary). We denote the boundary of U with respect to the inner metric by ∂eU = U \U, U and the distance to the boundary by e δ (x) = inf d (x,y) = inf d(x,y). U U y∈∂UeU y∈Uc For any open set V ⊂ U, let VdU denote the completion of V with respect to the metric d . We denote the boundary of V with respect to U by U ∂eV = VdU \V, e U and the part of boundary of V that lies in U by ∂UV = ∂UeV ∩U. In this work, we prove the boundary Harnack principle for the following class of domains. Definition 2.5 (Uniformandinner uniformdomains). Let U bea connected, opensubset of a length space (X,d). Let γ : [0,1] → U be a rectifiable, continuous curve in U. Let c ,C ∈ (0,∞). We say γ is a (c ,C )-uniform curve if U U U U δ (γ(t)) ≥ c min(d(γ(0),γ(t)),d(γ(1),γ(t))) U U for all t ∈ [0,1], and if L(γ) ≤ C d(γ(0),γ(1)). U The domain U is called (c ,C )-uniform domain if any two points in U can be joined by U U a (c ,C )-uniform curve. U U WesaythatU isanuniformdomainifU is(c ,C )-uniformdomainforsomeconstants U U c ,C ∈ (0,∞). We say that U is an inner uniform domain, if U is an uniform domain U U in (U,d ), where d is in the intrinsic metric corresponding to U. U U Another definition of uniform domains that appear in the literature is that of length uniform domains. Definition 2.6 (Length uniform and inner length uniform domains). Let U be a connected, open subset of a lengthspace (X,d). Let γ : [0,1] → U bea rectifiable, continuous curve in U. Let c ,C ∈ (0,∞). We say γ is a (c ,C )-length uniform curve if U U U U δ (γ(t)) ≥ c min L(γ ),L(γ ) U U [0,t] [t,1] (cid:16) (cid:17) (cid:12) (cid:12) for all t ∈ [0,1], and if (cid:12) (cid:12) L(γ) ≤ C d(γ(0),γ(1)). U 4 The domain U is called (c ,C )-length uniform domain if any two points in U can be U U joined by a (c ,C )-length uniform curve. U U We say that U is an length uniform domain if U is (c ,C )-length uniform domain U U for some constants c ,C ∈ (0,∞). U U We say that U is an inner length uniform domain, if U is a length uniform domain in (U,d ), where d is in the intrinsic metric corresponding to U. U U The following lemma extends the existence of inner uniform curves between any two points in U in Definition 2.6 to the existence of inner uniform curves between any two points in U. Lemma 2.7. Let (X,d) be a complete, locally compact, separable, length metric space. e Let U be a (c ,C )-inner uniform domain and let U denote the completion of U with U U respect to the inner metric d . Then for any two points x,y in (U,d ), there exists a U U (c ,C )-uniform curve in the d metric. e U U U e Proof. Let x,y ∈ U. There exist sequences (x ),(y ) in U such that x → y,y → y as n n n n n → ∞ in the d metric. Let γ : [0,1] → U,n ∈ N denote the (c ,C )-uniform curve U n U U in (U,d ) from x eto y with constant speed parametrization. By [BBI, Theorem 2.5.28], U n n the above curves can be viewed to be in the compact space B (x,2C d (x,y))dU for all U U U large enough n. By a version of Arzela-Ascoli theorem the desired inner uniform curve γ from x to y can be constructed as a sub-sequential limit of the sequence of inner uniform curves γ – see [BBI, Theorem 2.5.14]. (cid:3) n The following geometric property of a metric space (X,d) will play an important role in the paper. Definition 2.8 (Metric doubling property). We say that a metric space (X,d) satisfies the metric doubling property if there exists N > 0 such that any ball B(x,r) can be covered by at most N balls of radius r/2. A closely related notion is the volume doubling property. Definition 2.9 (Volume doubling property). We say that a Borel measure µ on a metric space (X,d) satisfies the volume doubling property, if there exist a constant C > 0 such D that µ(B(x,2r)) ≤ C µ(B(x,r)) for all x ∈ X and for all r > 0. D It is well known that volume doubling implies metric doubling. Further, by [LuS, Theorem 1] if a complete metric space (X,d) satisfies the metric doubling property then there exists a non-zero Borel measure µ satisfying the volume doubling property. Proposition 2.10. ([GyS, Proposition 3.3]) Let (X,d) be a complete, locally compact, separable, length metric space satisfying the metric doubling property. Then an open set U is a (resp. inner) length uniform domain if and only if U is (resp. inner) uniform domain. 5 Remark 2.11. The proof in [GyS] is inaccurate because the first displayed equation in [GyS, p. 82] which states B(x ,εr /2) ⊂ B(x,(1 + ε/2)r ) does not follow from r = j j j j min{r ,ρ(x,x )} ≤ ρ(x,x ). However this mistake can be easily fixed by following the j−1 j j proof of [MS, Lemma 2.7] more closely, using the parametrization of the curve as given in [MS]. Let U ⊂ X denote the closure of U in (X,d). Let p : (U,d ) → (U,d) denote the U natural projection map, that is p is the unique continuous map such that p restricted to U is the identity map on U. For any x ∈ U and for any ball De= B(p(x),r), let D′ denote the connected component of p−1(D∩U) containing x. We identify the subset D′∩p−1(U) of (U,d ) with the subset p(D′) ∩ U ofe(X,d) and simply denote it by D′ ∩ U. The U following lemma allows us to compare balls with respect to the d and d metrics. U e Lemma 2.12. Let (X,d) be a complete, length space satisfying the metric doubling property. Let U ⊂ X be a connected, open, (c ,C )-inner uniform domain. Then there exists U U C > 1 such that for all balls B(p(x),r/C ) with x ∈ U and r > 0, we have U U f BUe(x,r/CUf) ⊂ D′ ⊂ BUe(ex,r), where D′ the connected component of p−1(D ∩U) containing x. f Proof. See [LS, Lemma 3.7] where this is proved under the hypothesis of volume doubling, andnote thatthe argument onlyuses metric doubling. (Alternatively, a doubling measure exists by [LuS, Theorem 1], and one can then use [LS]). (cid:3) The following lemma shows that every point in an inner uniform domain is close to a point that is sufficiently far away from the boundary. Lemma 2.13. ([GyS, Lemma 3.20]) Let U be a (c ,C )-inner uniform domain in a U U length metric space (X,d). For every inner ball B = Be(x,r) with the property that U B 6= BUe(x,2r) there exists a point xr ∈ B with c r U d (x,x ) = r/4 and δ (x ) ≥ . U r U r 4 Proof. We recall the proof for convenience. Since Be(x,2r) 6= Be(x,r), there exists a U U point y ∈ Be(x,r) with d(x,y) = r/2. By Lemma 2.7, there exists γ : [0,1] → U a U (c ,C )-inner uniform curve joining x and y with γ(0) = x,γ(1) = y. By intermediate U U value theorem, there exists t ∈ (0,1)such that d (x,γ(t)) = r/4. Since γ is (c ,C )-inener U U U uniform, we obtain δ (γ(t)) ≥ c min(d (x,γ(t)),d (y,γ(t))) U U U U ≥ c min(d (x,γ(t)),d (x,y)−d (x,γ(t))) = c r/4. U U U U U Therefore x = γ(t) satisfies the desired properties. (cid:3) r 6 Lemma 2.14. Let U be a (c ,C )-inner uniform domain in a length metric space (X,d). U U If x,y ∈ U the there exists a (c ,C )-inner uniform curve γ connecting x and y with U U δ (z) ≥ 1c (δ (x)∧δ (y)) for all z ∈ γ. U 2 U U U Proof. Write t = δ (x)∧δ (y). Let γ be an inner uniform curve from x to y and z ∈ γ. U U d(z,x) ≤ 1t then δ (z) ≥ δ (x)−d(x,z) ≥ 1t, and the same bound holds if d(z,y) ≤ 1t. 2 U U 2 2 Finally if d(z,x)∧d(z,y) ≥ 1t then δ (z) ≥ 1c t. (cid:3) 2 U 2 U 3 Dirichlet space and Hunt process Let (X,d) be a locally compact, separable, metric space and let µ be a Radon measure with full support. Let (E,F) be a regular strongly local Dirichlet form on L2(X,µ) – see [FOT]. Recall that a Dirichlet form (E,F) is strongly local if E(f,g) = 0 for any f,g ∈ F with compact supports, such that f is constant in a neighbourhood of supp(g). We call (X.d,µ,E,F) a MMD space. Let L be the generator of (E,F) in L2(X,µ); that is L is a self-adjoint and non- positive-definite operator in L2(X,µ) with domain D(L) that is dense in F such that E(f,g) = −hLf,gi, for all f ∈ D(L) and for all g ∈ F; here h·,·i, is the inner product in L2(X,µ). The associated heat semigroup P = etL,t ≥ 0, t is a family of contractive, strongly continuous, Markovian, self-adjoint operators in L2(X,µ). We set E (f,g) = E(f,g)+hf,gi, ||f|| = E (f,f)1/2. (3.1) 1 E1 1 It is known that corresponding to a regular Dirichlet form, there exists an essentially unique Hunt process X = (X ,t ≥ 0,Px,x ∈ X). The relation between the Dirichlet form t (E,F) on L2(X,µ) and the associated Hunt process is given by the identity P f(x) = Exf(X ), t t for all f ∈ L∞(X,µ), for every t > 0, and for µ-almost all x ∈ X. We define capacities for (X,d,m,E,Fm) as follows. For a non-empty open subset D ⊂ X, let C (D) denote the space of all continuous functions with compact support in 0 D. Let F denote the closure of Fm ∩ C (D) with respect to the E (·,·)1/2 -norm. By D 0 1 A ⋐ D, we mean that the closure of A is a compact subset of D. For A ⋐ D we set Cap (A) = inf{E(f,f) : f ∈ F and f ≥ 1 in a neighbourhood of A}. (3.2) D D A statement depending on x ∈ B is said to hold quasi-everywhere on B (abbreviated as q.e. on B), if there exists a set N ⊂ B of zero capacity such that the statement if true for every x ∈ B \N. 7 For a Borel subset A ⊂ X, we denote by T := inf{t > 0 : X ∈ B}, τ := T = inf{t > 0 : X ∈/ U}. (3.3) B t B X\B t It is known that every function f ∈ F admits a quasi continuous version (cf. [FOT, Theorem 2.1.3]), and throughout this paper, we always assume that every f ∈ F is represented by its quasi-continuous version, which is unique up to a set of zero 1-capacity. Also associated with the Dirichlet form and f ∈ F is the energy measure dΓ(f,f). This is defined to be the unique measure such that for all bounded g ∈ F we have gdΓ(f,f) = 2E(f,fg)−E(f2,g). Z We have E(f,f) = dΓ(f,f). ZX Definition 3.1. For an open subset of U of X, we define the following function spaces associated with (E,F). F (U) = u ∈ L2 (U) : ∀ relatively compact V ⊂ U,∃u# ∈ F,u = u# µ-a.e. , loc loc V F(U) = (cid:8)u ∈ F (U) : |u|2dµ+ dΓ(u,u) < ∞ , (cid:12) (cid:9) loc (cid:12) (cid:26) ZU ZU (cid:27) F (U) = {u ∈ F(U) : The essential support of u is compact in U}, c F0(U) = the closure of F (U) in F for the norm E (u,u)1/2. c 1 For an open set U, another equivalent definition of F0(U) is given by F0(U) = {u ∈ F : u = 0 q.e. on X \U}, (3.4) where u is a quasi continuous version of u. We refer the reader to [FOT, Theorem e 4.4.3(i)] for a proof of this equivalence. By the equivalent definition, we can identify the F0(U) as a subset of L2(U,µ) where L2(U,µ) is identified with the subspace e {u ∈ L2(X,µ) : u = 0 µ-a.e. on X \U}. Definition 3.2. For an open set U ⊂ X, we define the Dirichlet-type Dirichlet form on U by D(ED) = F0(U) and ED(f,g) = E(f,g) for f,g ∈ F0(U). U U If (E,F) is a regular, strongly-local Dirichlet form on L2(X,µ) then (ED,F0(U)) is a U regular, strongly-local Dirichlet form on L2(U,µ). Moreover, (ED,F0(U)) is the Dirichlet U form of the process X killed upon exiting U. We write (PD,t ≥ 0) for the associated t semigroup, and call (PD) the semigroup of X killed on exiting D. See [CF, Theorem 3.3.8 t and Theorem 3.3.9] or [FOT, Theorem 4.4.3] for more details. For an open set U, we will often consider functions that vanish on a portion of the boundary of U, and therefore define the local spaces associated with (ED,F0(U)). U 8 Definition 3.3. Let V be an open subset of U, where U is an open subset of X. F0 (U,V) ={f ∈ L2 (V,µ) : ∀ open A ⊂ V relatively compact in U with loc loc d (A,U \V) > 0, ∃f♯ ∈ F0(U) : f♯ = f µ-a.e. on A}. U Roughly speaking, a function in F0 (U,V) vanishes along the portion of boundary loc given by ∂eV ∩∂eU. U U The extended Dirichlet space F0(U) is defined as the family of all measurable, almost e everywhere finite functions u such that there exists an approximating sequence (u ) ⊂ n F0(U) that isED-Cauchy andu = limu µ-almosteverywhere. If (ED,F0(U)) istransient U n U then F0(U) is a Hilbert space under the ED inner product, by [FOT, Lemma 1.5.5]. e U 4 Harmonic functions and the elliptic Harnack inequality 4.1 Harmonic functions We define harmonic functions for a strongly local, regular Dirichlet form (E,F) on L2(X,µ). Definition 4.1. Let U ⊂ X be open. A function u : U → R is harmonic on U, if u ∈ F (U) and for any function φ ∈ F (U) loc c E(u#,φ) = 0 where u# ∈ F is such that u# = u in the essential support of φ. Remark 4.2. (a) By the locality of (E,F), E(u#,φ) does not depend on the choice of u# in Definition 4.1. (b) If V ⊂ U, where U and V are open subsets of X and if u is harmonic in U, then the restriction u is harmonic in V. This again follows from the locality of (E,F). V (c) It is known t(cid:12)hat u ∈ L∞(U,µ) is harmonic in U if and only if it satisfies the following (cid:12) loc property: foreveryrelativelycompactopensubsetV ofU,t 7→ u(X )isauniformly t∧τV integrable Px-martingale for q.e. x ∈ V. (Here u is a quasi continuous version of u on V.) This equivalence between the weak solution formulation in Definition 4.1 and e the probabilistic formulation using martingales is given in [Che, Theorem 2.11]. e Definition 4.3. Let V ⊂ U be open. We say that a harmonic function u : V → R satisfies Dirichlet boundary conditions along the boundary ∂eU ∩VdU, if u ∈ F0 (U,V). U loc where VdU is the closure of V in (U,d ). U e e e 9 4.2 Elliptic Harnack inequality Definition 4.4 (Elliptic Harnack inequality). We say that (E,F) satisfies the elliptic Harnack inequality EHI, if there exist constants C < ∞ and δ ∈ (0,1) such that, for H any ball B(x,R) ⊂ X and any non-negative function u ∈ F (B(x,R)) that is harmonic loc on B(x,R), we have esssup u(z) ≤ C essinf u(z). (EHI) z∈B(x,δR) H z∈B(x,δR) We say that (E,F) satisfies the local EHI, denoted (EHI) , if there exists R ∈ (0,∞) loc 0 such that the (EHI) holds for all balls B(x,r) with r < R. An easy chaining argument show that if the EHI holds for some δ ∈ (0,1) then it holds for any other δ′ ∈ (0,1). Further, if the local EHI holds for some R then it holds (with 0 of course a different constant C ) for any other R ∈ (0,∞). H We recall the definition of Harnack chain – see [JK, Section 3]. Definition 4.5 (Harnack chain). Let U ( X be a connected open set. An M-nontangential ball in a domain U is a ball B(x,r) in U whose distance from ∂U is comparable to its radius r: Mr > d(B(x,r),∂U) > M−1r. Forx ,x ∈ U, aM-Harnack chain from x to x inU isasequence ofM-nontangential 1 2 1 2 balls such that the first ball contains x , the last contains x , and such that consecutive 1 2 balls intersect. Note that consecutive balls must have comparable radius. The number of balls in a Harnack chain is called the length of the Harnack chain. Remark 4.6. Assume that (E,F) satisfies the elliptic Harnack inequality. Then if u is a positive, continuous, harmonic function in U, then C−Nu(x ) < u(x ) < CNu(x ), (4.1) H 1 2 H 2 where N is the length of the minimal δ−1-Harnack chain between x , and x and C , δ 1 2 H are the constants in the EHI. For a domain D write N (x,y;M) for the length of the shortest M-Harnack chain in D D connecting x and y. Lemma 4.7. Let (X,d) be a locally compact, separable, length metric space that satisfies the metric doubling property. Let U ( X be a (c ,C )-inner uniform domain in (X,d). U U Then there exists C ∈ (0,∞), depending only on c , C and M, such that U U d (x,y) d (x,y) C−1log U +1 ≤ N (x,y;M) ≤ Clog U +1 +C D min(δ (x),δ (y)) min(δ (x),δ (y)) (cid:18) U U (cid:19) (cid:18) U U (cid:19) for all x,y ∈ U. Proof. See [GO, Equation (1.2) and Theorem 1.1] or [Aik15, Theorem 3.8 and 3.9]) for a similar statement for the quasi-hyperbolic metric on D; the result then follows by a comparison between the quasi-hyperbolic metric and the length of Harnack chains as in [Aik01, pp. 127]. (cid:3) 10

Boundary Harnack principle and elliptic Harnack inequality

Similar Boundary Harnack principle and elliptic Harnack inequality