The Dirichlet Heat Kernel in Inner Uniform Domains in Fractal-Type Spaces

This paper proves two-sided estimates for the Dirichlet heat kernel on inner uniform domains in metric measure Dirichlet spaces satisfying the volume doubling condition, the Poincaré inequality, and a cutoff Sobolev inequality. More generally, we obtain local upper and lower bounds for the Dirichlet heat kernel on locally inner uniform domains under local geometric assumptions on the underlying space.


Introduction
This paper continues the study of the Dirichlet heat kernel by the Doob's transform technique. P. Gyrya and L. Saloff-Coste [20] proved that the Doob's h-transform of the Dirichlet heat kernel p D U by a harmonic profile h on an unbounded domain U (i.e. h is harmonic on U and satisfies Dirichlet boundary condition), p h (t, x, y) := p D U (t, x, y) h(x)h(y) , satisfies two-sided Gaussian bounds provided that U is an inner uniform domain in a metric measure Dirichlet space that admits a global heat kernel satisfying Gaussian bounds. This yields two-sided bounds for the Dirichlet heat kernel in terms of the harmonic profile h, where d U is the intrinsic length metric of the domain U , and (r) = r 2 is the appropriate time-space scaling for a diffusion that shows classical Gaussian behavior.
Similarly, for the case of bounded inner uniform domains, L. Saloff-Coste and the author [27] obtained two-sided bounds for the Dirichlet heat kernel by proving Gaussian bounds for the kernel p φ (t, x, y) := p D U (t, x, y)e tλ φ(x)φ(y) , where φ is the ground state and λ is the lowest Dirichlet eigenvalue for the domain U . The Dirichlet heat kernel estimates obtained by this method imply the intrinsic ultracontractivity of the infinitesimal generator of the heat semigroup. In particular, they refine the estimates of, e.g., [5,14] in the special case of inner uniform domains. In [27], Dirichlet heat kernels on locally inner uniform domains, and non-symmetric Dirichlet heat kernels were studied as well. The aim of this paper is to prove two-sided estimates for Dirichlet heat kernels on bounded inner uniform domains and on locally inner uniform domains in fractal-type spaces such as the Sierpinski gasket where a global heat kernel exists and satisfies two-sided sub-Gaussian estimates. The main results of this paper are stated in Section 3. As special cases, we recover the two-sided estimates for Dirichlet heat kernels of [20] and [27] (Gaussian case), and [22] (sub-Gaussian case). The setting of this paper is that of a symmetric strongly local regular Dirichlet space satisfying volume doubling, reverse volume doubling, Poincaré inquality and a cutoff Sobolev condition. Characterizations of sub-Gaussian heat kernel estimates by these conditions, or by the parabolic Harnack inequality, are given in, e.g., [10,11,18,19,21]. The hypotheses on the underlying metric measure Dirichlet space are described in Section 1 where we also recall the definition of inner uniform domains and how it relates to the boundary Harnack principle. If we assume volume doubling and Poincaré inequality only locally, then we still get estimates for the Dirichlet heat kernel p D U locally near two points x and y and for small time t. These local estimates hold on any locally inner uniform domain.
In Section 2, we consider Doob's h-transform and show that volume doubling, Poincaré inquality and the cutoff Sobolev inequality on annuli are preserved under Doob's htransform.
In Section 4 we consider examples of inner uniform domains in the Sierpinski gasket. These are constructed by taking a line that cuts through the gasket as boundary of the domain. For horizontal or vertical lines, computations of harmonic functions and their boundary decay is known. We quote some of these examples. As can be seen from our main results, the boundary decay rate of the Dirichlet heat kernel at any fixed boundary point is the same as the boundary decay rate of the ground state φ, which in turn is the same as the boundary decay rate of any function that is positive harmonic on some portion of the domain near that boundary point and vanishes along the boundary of the domain.
The Dirichlet form (E, F) admits an energy measure d (u, v), for each u, v ∈ F. In particular, E(u, v) = X d (u, v). We will tacitly use the well-known properties of the energy measures such as chain rule and Cauchy-Schwarz type inequality; these are presented in [17, (3.2.27) (u, u) is denoted as μ u , and in [13].
For an open subset U ⊂ X, define where L 2 loc (U, μ) is the space of functions that are L 2 (μ)-integrable over any open relatively open relatively compact and f , g are functions in F such that f = f , g = g μ-a.e. on A. Indeed, this construction is justified by [13,Theorem 4.3.10(ii)] and relies on the fact that f | A and f | A have quasi-continuous modificationsf andf , respectively, which agree on A. Define where the F-norm is defined as is a Dirichlet form and the functions in its domain F 0 (U ) satisfy the Dirichlet boundary condition. Let (L D U , D(L D U )) be the infinitesimal generator and P D U,t , t > 0, be the semigroup associated with (E D U , F 0 (U )).

Volume doubling, Poincaré and cutoff Sobolev inequalities
In this section we recall geometric conditions that will be presumed in the main results. It is well known that these are satisfied on many self-similar fractals; we refer to [6-8, 10, 16, 23]. Let Y ⊂ X be open and R 0 ∈ (0, ∞]. We will assume that any ball B(x, R) = {y ∈ X : d(x, y) < R} with R ∈ (0, R 0 ) is relatively compact whenever B(x, 2R) ⊂ Y . In alignment with the notation in [27], we will call this condition A2-Y , or simply A2 when Y = X.
where V (x, R) = μ(B(x, R)) denotes the volume of the ball B(x, R).
(ii) The reverse volume doubling property (RVD) holds on Y up to scale R 0 if there are constants C RVD and ν 0 ∈ [1, ∞) such that for any ball B(x, 2R) ⊂ Y with R ∈ (0, R 0 ) and X \ B(x, R) = ∅, and for all s ∈ (0, R), y ∈ B(x, R).
Definition 1. 4 We say that CSA( ) holds on Y up to scale R 0 if there exists a constant C 0 > 0 such that for some ∈ (0, 1), any B(x, 2R) ⊂ Y with R ∈ (0, R 0 ), and any r ∈ (0, R], there exists a cutoff function φ for B(x, R) in B(x, R + r) which satisfies where If CSA( ) holds on X up to scale R 0 = ∞ for fixed = 1 8 , then we simply say that CSA( ) holds. This global condition was introduced in [4] and named the cutoff Sobolev inequality on annuli. Remark 1.5 (i) If A2-Y , RVD and CSA( ) hold on Y up to scale R 0 for some ∈ (0, 1), then it holds for all ∈ (0, 1). See [24,Lemma 2.3]. (ii) If A2-X, RVD, VD and PI( ) hold on X up to scale R 0 = ∞, then CSA( ) is equivalent to the generalized capacity condition introduced in [18]. In general, the generalized capacity condition is weaker than CSA( ) since it allows the cutoff function φ to depend on the function f .
We will drop the words "up to scale R 0 " in any of the above conditions when R 0 = ∞.
We say that A2, VD, RVD, or PI( ) holds locally on X if every point x ∈ X has a neighborhood Y x = B(x, 4r x ) where A2-Y x , VD, RVD, or PI( ), respectively, holds up to scale R 0 = 4r x ≤ 1 2 diam(X). Here, diam(X) denotes the diameter of (X, d).

Harnack Inequalities
Let I be an open time interval and U ⊂ X an open. For the definition of a local weak solution to the heat equation on Q = I × U , we will use the following function spaces.
Recall that F is the domain of the Dirichlet form (E, F). We denote by F the dual space of F. Let L 2 (I → F) be the Hilbert space of those functions v : Similarly, we define the space L 2 (I → F ). Observe that we can identify L 2 (I → F ) with the dual space of L 2 (I → F), and we may write v(t, x) for (v(t))(x).
Let W 1 (I → F ) ⊂ L 2 (I → F ) be the Hilbert space of those functions v : I → F who have a distributional time-derivative that can be represented by a function in L 2 (I → F ). Identifying L 2 (I → L 2 (X)) with its dual space, we set Let For other notions of a local weak solution which are equivalent to the one given above see, e.g., [26] and references therein. The existence and uniqueness of weak solutions is known from PDE literature [29,Section 4 in Chapter 3] and [35].
If λ = 0, we simply say that u is harmonic or E-harmonic.
Harmonic functions are time-independent weak solutions of the heat equation. Therefore, the following elliptic Harnack inequality is an immediate consequence of the parabolic Harnack inequality PHI( ).
The Dirichlet form (E, F) satisfies the elliptic Harnack inequality (EHI) on Y up to scale R 0 if for any δ ∈ (0, 1) there is a constant C EHI > 0 such that, for any ball B(x, 2r) B(x, 4r) ⊂ Y with r ∈ (0, R 0 /4), and any function u ∈ F that is non-negative and harmonic on B(x, r), we have If U is connected, then U admits an inner metric The inner metric d U is indeed finite because U is connected and (X, d) is a length metric space by assumption. Let U be the completion of U with respect to d U , and let ∂ U U = U \U be the boundary.
Note that F 0 loc (U ) is the same as F 0 loc (U, U ).
We say that a harmonic function u : V → R satisfies Dirichlet boundary condition along ∂U if u ∈ F 0 loc (U, V ). and The domain U is called a (c, C)-inner uniform domain if any two points in U can be joined by a (c, C)-inner uniform path in U .
Fix c ∈ (0, 1) and C ∈ (1, ∞). For ξ ∈ ∂ U U , let R ξ be the largest radius so that Let C 0 be the space of continuous functions f : X → R that vanish at infinity. It remains to show that lim t→0 P t f − f ∞ = 0 for any f ∈ C 0 . The proof in [25, Proposition 3.2] uses a conservativeness argument which does not apply here. However, the assertion can be proved by approximating P t f by P t g for some continuous g. Choosing g to be constant on a neighborhood U of an arbitrary point x 0 ∈ X, it is possible to extend P t g to a local weak solution on the whole time interval (−∞, ∞) by setting it equal to  8 9 R ξ ) be small enough so that A2, VD, PI( ) and CSA( ) hold on Thus, the reasoning of [25, Section 3.2, Section 4, Section 5] goes through under the weaker assumptions of Theorem 1.12, proving the boundary Harnack principle.

Proposition 1.11 Suppose (X, d) is geodesic, A2, CSA( ) hold globally on X, and VD, PI( ) hold locally on X. Then the Markovian transition function (P t ) t>0 associated with (E, F) has the Feller-type property
We need the following two known results for the proof of our main results, and restate them here for the convenience of the reader.   Let p : U → U be the natural projection into the closure U of U in (X, d), namely the unique continuous map such that p| U is the identity map on U . For x ∈ U and D = The proof of the next lemma is given in [28, Lemma 3.7, Remark 3.8(ii)].
The constant C U depends only on the volume doubling constant C VD and on the inner uniformity constants c, C. It does not depend on x or r.

Doob's h-transform Technique
For this section, we let (X, d, μ, E, F) be a metric measure Dirichlet space as in Section 1.1. In addition, we assume that d is a length metric and A2 holds. We fix a domain U ⊂ X and a subdomain W ⊂ U . Let R 0 ∈ (0, ∞].

Doob's h -transform
for the norm It follows easily from the definition that (E h , F h ) is a symmetric strongly local regular Dirichlet form on For a positive continuous function h on W , let Then the linear subspace and a Hilbert space with inner product The following lemma is proved in [27,Lemma 6.5]. See also [20, Proof of Proposition 5.7].

Lemma 2.2 If h is a positive continuous function in F loc (W ), then
and this space is dense in the Hilbert space for any continuous function ψ ∈ F so that ψ| V has compact support in V and ψ| V extends continuously to V .
Proof We follow the line of reasoning in [20, Proof of Lemma 2.36].
We first prove the forward implication. Let ψ ∈ F so that ψ| V has compact support in V and ψ| V is continuous and extends continuously to V . For a bounded function f ∈ F 0 loc (U, V ) and any compact set By the dominated convergence theorem, we obtain that (ψf n ) is a Cauchy sequence in F 0 (V ). Hence, (ψf n ) converges to ψ| V f ∈ F 0 (V ) by the closedness of (E, F).
To prove the converse, let f be a bounded function on V . Let K be a compact set in V . Take a continuous function ψ ∈ F that takes constant value 1 on K ∩ V and so that ψ| V has compact support in V . As ψ| V f (x) = f (x) for μ-a.e. x ∈ K ∩ V , if we know that ψ| V f is in F 0 (V ) (for an arbitrary compact set K ⊂ V ), then we must have f ∈ F 0 loc (U, V ). A proof of the following proposition in the special case when U = W is an unbounded inner uniform domain and R 0 = ∞ will be given in [22].

Proposition 2.4 For a positive continuous function
Consider the collection C K,n of functions in C c W ∩ F h whose absolute value is at most 1 n outside K. Restricting the functions in C K,n to K, we obtain a collection of functions in C(K) which separates the points of K and contains the function that takes constant value 1 n on K; this will be proved below. By the Stone-Weierstrass theorem, e.g. [34,Theorem 44.5], there exists a sequence (f n,m ) m ⊆ C K,n such that sup K |f n,m − u| < 1 m . The diagonal sequence (f n,n ) n converges to u uniformly on W .
It remains to verify the required properties of C K,n . We first show that there exists a function in C K,n which takes constant value 1 n on K. Indeed, since (E, F) is regular, there exists a continuous function φ ∈ F which takes value 1 n on an open neighborhood of p(K), and so that φ| W has compact support in W . Then φ| W h ∈ F 0 (W ) by Lemma 2.3, consequently φ| W ∈ H −1 (F 0 (W )). By Lemma 2.2, there exists a sequence (φ n ) ⊂ F c (W ) ∩ L ∞ (W, h 2 dμ) such that hφ n → hφ| W in F 0 (W ). We may assume that (hφ n ) are uniformly bounded. Passing to a subsequence, we may assume that hφ n converges to hφ| W quasi-everywhere. In particular, φ n converges to φ| W quasi-everywhere. By the Leibniz rule and Cauchy-Schwarz inequality, Squaring each side and rearranging, As explained above, the right hand side converges as n → ∞. Passing again to a subsequence, this shows that the subsequence (φ n ) converges in F h , and its limit must be φ| W .
In particular, φ| W ∈ F h . By construction, φ| W extends continuously to W . Next, we show that C K,n separates any two distinct points x, y ∈ K. This property clearly holds if at least one of the points is in K ∩ W . For two distinct points x, y ∈ K \ W , their inner distance d U (x, y) is positive. Let 0 < d < d U (x, y)/2 small. By regularity of (E, F), we can find a function f ∈ C c (X) ∩ F so that f (p(x)) = 1 and f = 0 outside B(p(x), d). (B(p(x), d)) as x, and φ(z) = 0 otherwise, we obtain that φ(x) = 1 and φ(y) = 0. When d is small enough, φ is in C c (W ). We have φ| W h ∈ F 0 (W ) by Lemma 2.3. By the same reasoning as above, it follows that φ| W ∈ F h .

Harmonic profiles
Let W ⊂ U ⊂ X be as in the previous subsection and λ ∈ R.
For any harmonic profile h, we set h(x) := 0 for all x ∈ X \ U . Harmonic profiles can be constructed using Green functions and the boundary Harnack principle. In [20, Theorem 4.16] the construction of an E-harmonic profile is given for unbounded inner uniform domains in a Harnack-type Dirichlet space. In fact, in this case the E-harmonic profile is unique up to multiplication by a scalar. The existence and uniqueness of an E-harmonic profile for unbounded inner uniform domains in a metric measure Dirichlet space with length metric and satisfying VD, PI( ), CSA( ) will be given in [22].
Bounded inner uniform domains do not have an E-harmonic profile due to the maximum principle. Instead one may use the ground state which is a an E − λ-harmonic profile when λ is the principle Dirichlet eigenvalue.
We will use (U, W )-harmonic profiles when the domain is bounded inner uniform or only locally inner uniform (cf. Definition 1.10). For instance a Green function on U with pole in U \ W would be a (U, W )-harmonic profile; see also [27,Proposition 5.9].
The next lemma is a special case of [27, Lemma 6.6], we include the proof for the convenience of the reader. Lemma 2.6 Let λ ∈ R and let h be a (U, W )-profile for E − λ(·, ·). Then Proof We first show that Eq. 8 holds. Indeed, this easily follows from the E −λ-harmonicity Applying again the E-harmonicity of h, we see that (hf n ) is a Cauchy sequence in (F 0 (W ), · F ) with L 2 -limit hf . The closedness of (E D W , F 0 (W )) now yields that hf ∈ F 0 (W ) and hf n → hf in (F 0 (W ), · F ). Together with Lemma 2.2, we see that the two spaces H −1 (F 0 (W )) and F h coincide.

Doob's h -transform on Locally Inner Uniform Domains
Assume that (i) U is locally (c u , C u )-inner uniform up to scale R 0 near W (in the sense of Definition 1.10), (ii) A2 and CSA( ) hold on X, (iil) VD, RVD, and PI( ) hold locally on {x ∈ X : d(x, w) ≤ R 0 for some w ∈ W } up to scale R 0 . A proof of the following proposition in the special case when U = W is an unbounded inner uniform domain and R 0 = ∞ was obtained in collaboration of the author with N. Kajino, and will be given in [22].    R). It suffices to show (10) for bounded f ∈ F h . Indeed, we can then apply (10) to f n := (f ∧ n) ∨ (−n) and, letting n → ∞ and using Fatou's lemma, obtain (10) for general f ∈ F h . Note that A U ⊂ A∩U . Due to the Dirichlet boundary condition of h, the energy measure h 2 d does not charge X \ U . The positivity of h 2 d (ψ, ψ) and strong locality then yield We will apply (9) to the right side of the above estimate. We also use the Leibniz rule for the energy measure, strong locality and the fact that g D is constant on each connected component of the support of φ, and the Cauchy-Schwarz inequality. We obtain R). We let η → 0 and get the above estimate with At this point we can conclude that the cutoff Sobolev inequality holds on balls for the inner metric (rather than on annuli). Indeed, by the product rule and Cauchy-Schwarz inequality for the energy measure and the E-harmonicity of h, we have hence the cutoff Sobolev inequality on balls for the inner metric follows by putting the above estimate into Eq. 11 and rearranging terms. Let δ = r 100 . For x i ∈ A U , let φ i be a cutoff function for B U (x i , τ δ) in B U (x i , δ) which satisfies the cutoff Sobolev inequality on balls for a given = 1 32 . The parameter τ ∈ (0, 1) is provided by the cutoff Sobolev inequality on balls for the inner metric, which we proved above. We choose the points x i in such a way A U ⊂ i B U (x i , τ δ) and no more than N of the balls B U (x i , δ) intersect in any given point, where N depends only on the constants in VD and RVD. Observe that i φ 2 i ≥ 1 on A U . By the E-harmonicity of h, the Cauchy-Schwarz inequality and the product rule for the energy measure, Rearranging, using the fact that i φ 2 i ≤ N , and applying the cutoff Sobolev inequality for each of the cutoff functions φ i , as well as the Cauchy-Schwarz inequality and the product rule for the energy measure, Because ψ = 1 on B U (x, R), the energy measure d (ψ, ψ) does not charge B U (x, R) by strong locality. Choosing = 1 32 and rearranging once again, Combining this with Eq. 11, (r) . Choosing c 1 > 0 sufficiently small, we obtain the desired estimate (10).  − λ(·, ·)). Then there are constants a 0 , A 0 ∈ (0, ∞) such that the Dirichlet space (W , d U ,ĥ 2 dμ, Eĥ, Fĥ) satisfies VD, PI( ), CSA( ), PHI( ) locally on any ball B U (x, a 0 R) up to scale a 0 R, provided that B U (x, A 0 R) ⊂ W and a 0 R ∈ (0, R 0 ). The constants in these conditions depend on an upper bound for |λ| (R 0 ).

Proposition 2.10 Let λ ∈ R and letĥ be a (U, W )-profile for (E
Moreover, if h is a (U, W )-profile for E, then , for all y, z ∈ B U (x, a 0 R).

Main Results: Estimates for Symmetric Dirichlet Heat Kernels
Let (X, d, μ, E, F) be a (symmetric, strongly local, regular) metric measure Dirichlet space as in Section 1.1. We fix a domain U ⊂ X and a subdomain W ⊂ U . Let Then there are constants a 0 ∈ (0, 1) and A 1 , a 2 , a 3 , A 4 ∈ (0, ∞) such that the Dirichlet heat kernel p D U exists and satisfies the upper bound for all t ∈ (0, T ), ξ ∈ B U (x, a 0 R x ), ζ ∈ B U (y, a 0 R y ), and the near-diagonal lower bound for all t ∈ (0, T ), ξ ∈ B U (x, a 0 R x ), ζ ∈ B U (y, a 0 R y ) whenever d(x, y) ≤ R x /4. Here, Remark 3.2 A full lower bound (for any x, y ∈ X) can be obtained provided that (i) -(iv) hold for all x ∈ X up to a fixed scale R x = R 0 . This follows from the near-diagonal lower bound and from applying PHI( ) repeatedly along a Harnack chain of balls connecting x to y. This argument uses the fact that d U is a length metric.
In the special case when U is unbounded and R 0 = ∞, the global hypotheses A2, VD, RVD, PI( ), CSA( ) can equivalently be replaced by A2, VD, RVD and two-sided estimates for the global heat kernel on X thanks to [18,Theorem 1.2]. The principle Dirichlet eigenvalue λ equals zero in this case. The global Dirichlet heat kernel estimates in this special case will be the main result of the forthcoming paper [22] by N. Kajino. For bounded domains, we have the following result.
for all t ∈ (0, (R 0 )), x, y ∈ U , and  d(x, y) is large). Though the constants initially depend also on λ (R 0 ), this quantity can be estimated in terms of the other constants because here λ is the eigenvalue. Indeed, it suffices to apply the Courant-Fisher min-max principle and use the cutoff function given by CSA( ) as test function.
We have the following two corollaries. We omit the proofs since they are almost identical to the proofs of [27,Corollary 7.10 for some constant A 4 ∈ (0, ∞). Further, a 3 ≤ w ≤ A 3 , and where a 3 , A 3 , R are as in Theorem 3.3.

Sierpinski Gasket
Consider the standard Sierpinski gasket S. For the construction of the symmetric strongly local regular Dirichlet form associated with the Laplacian on the standard Sierpinski gasket S, see e.g., [3]. Let F n be the finite set of vertices in the graph approximation S n of the Sierpinski gasket at level n. We set F 0 = {s 1 , s 2 , s 3 }. Proof Without loss of generality we assume that S is scaled to have side length 1.
Suppose condition (C) is satisfied. Pick two arbitrary points x, y ∈ U . Let m be the largest integer such that x and y are contained in the same cell of side length 2 −m . Then x and y lie in different cells of side length 2 −m−1 . Therefore, any path from x to y in (U, d U ) must pass through at least one vertex V ∈ F m+1 ∩ U . Since L is a straight line, there exists a geodesic from x to V that lies in U , and as we move along this path the distance to the line is an affine function, so the geodesic is a c-John path with constant c depending only on c 0 . Similarly, we construct a c-John path from y to a vertex V ∈ F m+1 ∩U . If V = V , then the concatenation of the two John paths yields an inner uniform path from x to y. If V = V , then we include the edge from V to V in our path from x to y. This edge is adjacent to a hole whose side length is comparable to 2 −m−1 . By condition (C), the line stays away from this edge at scale 2 −m−1 . Thus, the path we constructed is indeed inner uniform.
Suppose condition (C) is not satisfied. Then, for any positive integer n, we can find two points x, y ∈ U on different edges of length 2 −m meeting at a vertex V such that d U (x, y)/2 −m < 1 n is small while d U (x, y)/d(V , L) > n is large. Thus, any path connecting x to y in U that satisfies (7) violates the banana condition (6).
2 for any integer n ≥ 2. It is well known that the semigroup associated with the Laplacian on the Sierpinski gasket admits a heat kernel p t (x, y) which satisfies the two-sided bounds where μ is the Hausdorff measure, d w = log 5/ log 2 and d s /2 = log 3/ log 5. See [6]. The heat kernel estimates of Theorem 3.3 apply to the Dirichlet heat kernel on the domains described in Proposition 4.1. It is a consequence of the main result of this paper, Theorem 3.3, that the Dirichlet heat kernel has the same boundary decay rate as any positive harmonic function that vanishes on a portion of the boundary.
The following examples illustrate how to compute boundary decay rates of harmonic functions. These and more general computations are essentially well known. The new contribution of the present paper is that the Dirichlet heat kernel shares the same boundary decay rate as the harmonic functions h studied below. Indeed, the Dirichlet heat kernel estimates of Theorem 3.3 imply that the Dirichlet heat kernel and the ground state φ share the same boundary decay rate. However, φ and h share the same boundary decay rate due to a version of the boundary Harnack principle: See [27, Proof of Theorem 6.12] which, thanks to [24,25], generalizes to the present setting of fractal spaces.
The first example is a well-known computation (e.g., [3, Example 3.2.6] or [37, Figure  2  Then there exists a unique non-negative continuous function h on S that is harmonic on U and satisfies the given boundary condition. Figure 1 shows the values a n , b n , c n that the For instance, the sequence (b n ) describes the boundary decay of h when approaching the boundary point s 3 (in the right corner of the Sierpinski gasket) from the left.
For horizontal lines, boundary value problems are studied in [30,32,36]. The harmonic function in the following example is computed in [   Harmonic functions on domains to the left of a vertical line are studied in [37], as well as in [12].

Sierpinski Carpet
In the Sierpinski carpet, inner uniform domains can be constructed by the same token as in Proposition 4.1. It is well known that the volume doubling property, the Poincaré inequality and the cutoff Sobolev inequality are all satisfied by the Dirichlet form associated with the Laplacian on the Sierpinski carpet. Hence, if U satisfies condition (C'), then Theorem 3.3 implies that the Dirichlet heat kernel exists on U and satisfies the two-sided bounds provided by the theorem.
If condition (C') is not satisfied, then U is a John domain and its center can be chosen among those of the four corners of the Sierpinski carpet that lie in U .
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