Mem. Fac. Sci. Eng. Shimane Univ.
Series B: Mathematical Science
31 (1998), pp. 1–7

SEMIBOUNDED PERTURBATION OF
GREEN FUNCTION IN AN NTA DOMAIN

Hiroaki Aikawa(Received December 10, 1997)
PAbstract. Let L = − i,j ∂i (aij ∂j ) be a Hölder continuous uniformly elliptic
operator on an NTA domain and let LV = L + V be a generalized Schrödinger
operator for a nonnegative function V . Let G and GV be the Green function for theDirichlet problem with respect to L and LV , respectively. In case V is a functionof the distance to the boundary, a necessary and suﬃcient condition for G and GV
to have comparable decay near the boundary is given. This is based on an integral
identity of L -harmonic functions and integral inequalities of L -superharmonic andL -subharmonic functions, which may be of independent interest.

∑1. Introduction

Let L = − i,j ∂i (aij ∂j ) be a smooth uniformly elliptic operator on a domain
D ⊂ Rn , n ≥ 2, such that aij (x) = aji (x) is Hölder continuous on D andΛ−1 |ξ|2 ≤ A[ξ] ≤ Λ|ξ|2for all ξ ∈ Rn ,∑nwhere A[ξ] =i,j aij ξi ξj and Λ > 1 is a constant independent of ξ ∈ R andx ∈ D. We also consider the Schrödinger operator LV = L + V for a nonnegative
function V on D. Let G and GV be the Green function for the Dirichlet problem
in D with respect to L and LV , respectively. This means that for any y ∈ D,G(·, y) and GV (·, y) vanish on ∂D and L G(·, y) = LV GV (·, y) = δy , where δy isthe Dirac measure at y. Since L and LV are self adjoint, it follows that G andGV are symmetric.
We are interested in the relationship between G and GV . It is easy to see that
GV (x, y) ≤ G(x, y)

for x, y ∈ D.

1991 Mathematics Subject Classiﬁcation. 31B05, 31C35, 35B20, 35J10.
Key words and phrases. Semibounded perturbation, Schrödinger equation, Green function,
NTA domain, Martin boundary, Coarea formula.
1

2HIROAKI AIKAWA

Let x0 ∈ D be ﬁxed. We say that V is semibounded perturbation of LV if
G(x0 , y) ≤ CGV (x0 , y)for y ∈ Dwith C > 1 independent of y ∈ D (see [13, §3]). The main purpose of this paper is togive a necessary and suﬃcient condition for V of a certain form to be semibounded
perturbation in the case when D is an NTA domain (see [10]). By the symbol
C we denote an absolute positive constant whose value is unimportant and maychange from line to line. We shall say that two positive functions f1 and f2 arecomparable, written f1 ≈ f2 , if and only if there exists a constant C ≥ 1 such thatC −1 f1 ≤ f2 ≤ Cf1 . The main result of this paper is the following.
Theorem 1. Let D be an NTA domain. Let v(r) be a locally bounded nonnegativefunction for 0 < r ≤ r0 = diam(D) such that v(r) ≈ v(2r) and r2 v(r) is nondecreasing for small r > 0. Set V (x) = v(δ(x)) with δ(x) = dist(x, ∂D). Then V issemibounded perturbation of LV if and only if∫rv(r)dr < ∞.
(1)
0It is known that the Martin boundary of an NTA domain with respect to L is
the Euclidean boundary and every boundary point is minimal ([10, Theorem 5.9and p.158]). Hence, in view of [14, Theorem 2], we have the following corollary
immediately.
Corollary 1. Let D be an NTA domain and let v and V be as above. If (1) holds,
then the Martin boundary of D with respect to LV is the Euclidean boundary of Dand every boundary point is minimal.
When L = −∆ and D is smooth, Suzuki [14] showed the suﬃciency part of
the above theorem and we gave an alternative proof as well as the necessity part([4, Theorems 2 and 3] see also [3]). Both proofs use the smoothness of D anddo not apply to an NTA domain. Instead, we shall employ the following integral
identity of L -harmonic functions and integral inequalities of L -superharmonicand L -subharmonic functions, which may be of independent interest. These aregeneralizations of our previous results [1] (see also [2] and [5, pp.171–182]), which
is based on the coarea formula.
Theorem 2. Let D be a regular domain. Let ϕ(t) be a nonnegative function fort > 0. Let x0 ∈ D and g(x) = G(x0 , x). Suppose 0 ≤ a < b ≤ ∞. Then thefollowing statements hold:(i) If h is L -harmonic on D, then∫ b∫ϕ(t)dt.
h(x)ϕ(g(x))A[∇g(x)]dx = h(x0 ){x∈D:a<g(x)<b}a

SEMIBOUNDED PERTURBATION OF GREEN FUNCTION

3(ii) If u is L -superharmonic on D, then∫∫ bu(x)ϕ(g(x))A[∇g(x)]dx ≤ u(x0 )ϕ(t)dt.
{x∈D:a<g(x)<b}a(iii) If s is L -subharmonic on D, then∫
∫ bs(x)ϕ(g(x))A[∇g(x)]dx ≥ s(x0 )ϕ(t)dt.
{x∈D:a<g(x)<b}

aWe shall estimate |∇g(x)| to obtain some corollaries of Theorem 2, which will
be needed for the proof of Theorem 1. The upper estimate is easy. The lower
estimate is more diﬃcult. If D is an NTA domain, then |∇g(x)| can be estimatedby g(x)/δ(x) in a certain sense with the help of the boundary Harnack principle([10, Theorem 5.1 and p.138]). For details see Section 2.
For the proof of Theorem 1 we also need the decay estimate of g(x) near theboundary. We ﬁnd 0 < β ≤ 1 ≤ α < ∞ such that(2)C −1 δ(x)α ≤ g(x) ≤ Cδ(x)βfor x ∈ D close to ∂D.

The above inequalities are rather easy. The ﬁrst inequality actually holds for a Johndomain (see e.g. [9, p.185 and (2.6)]) and the second does for a domain satisfying
the capacity density condition (see e.g. [7]). In general, β < 1 < α. Only in thecase when D is smooth, we have α = β = 1, which was essential in [4, Theorems 2and 3]. Surprisingly, the weak estimate (2) is suﬃcient for the proof of Theorem 1.
Recently, Ancona [8] studied Green functions for elliptic operators on manifolds
or domains. By using a completely diﬀerent method, he proved a better result
than the suﬃciency part of Theorem 1. He proved that under the condition (1), Vbecomes bounded purtabation of LV , i.e., G(x, y) ≈ GV (x, y) for all x, y ∈ D.
2. Proof of Theorem 2 and corollariesi
Proof of Theorem 2. The proof is essentially the same as in [1]. We shall prove only(i), since the remaining can be proved in the same fashion. Let h be L -harmonicon D. Let Dt = {x ∈ D : g(x) > t} for t > 0. Observe that Dt is a relatively
compact subset of D. We see that the Green function Gt for Dt with respect to L
satisﬁes that Gt (x0 , x) = g(x) − t. By the Sard theorem (see e.g. [11, Corollary onp.35]), ∂Dt is a smooth surface for a.e. t > 0. For such t we see that the outward
normal n of ∂Dt is given by −∇g/|∇g|. On the other hand the Poisson integral
formula for Dt with respect to L shows that∫h(x0 ) = −h(x)(∇g(x), A(x)n(x))dσ(x),∂Dt4where A(x)n(x) = (HenceHIROAKI AIKAWA∑

j aij nj ) is the conormal vector (see [12, (10.4) on p.21]).

∫h(x0 ) =h(x)∂DtA[∇g(x)]dσ(x).
|∇g(x)|

By the coarea formula (see e.g. [11, pp.37–39])
∫ b ∫
∫ b
A[∇g(x)]dt
h(x)ϕ(g(x))ϕ(t)dt =h(x0 )dσ(x)|∇g(x)|a∂Dta∫
h(x)ϕ(g(x))A[∇g(x)]dx.
={x∈D:a<g(x)<b}

Thus the required identity holds.
Let us estimate ∇g(x) and give some corollaries of Theorem 2, which will beneeded for the proof of Theorem 1. Since g(x) is positive and L -harmonic near∂D, it follows that |∇g(x)| ≤ Cg(x)/δ(x). Since A = (aij ) is uniformly elliptic, wehave the following corollary from the above proof.
Corollary 2. Let 0 ≤ a < b ≤ ∞. Suppose that s is a positive L -subharmonic
function on {x ∈ D : g(x) > a}. Then∫∫ bg(x)2ϕ(t)dt ≤ C
s(x)ϕ(g(x))
s(x0 )dx,
δ(x)2a{x∈D:a<g(x)<b}
where C is independent of s, a and b.
The inequality |∇g(x)| ≥ Cg(x)/δ(x) does not hold in a pointwise sense. However, it holds in a certain weak sense for an NTA domain. In [1] this was observedfor L = −∆. The key ingredient was the boundary Harnack principle. For a general uniformly elliptic operator L the boundary Harnack principle is available also(see [10, Theorem 5.1 and p.138]) and hence the same proof as in [1] works for thepresent case. We have the following corollary.
Corollary 3. Let D be an NTA domain. Let ϕ(t) ≈ ϕ(2t) for t > 0 and let0 ≤ a < b < ∞. Suppose that u is a positive L -superharmonic function on D.
Then∫∫ bg(x)2u(x)ϕ(g(x))
dx ≤ Cu(x0 )ϕ(t)dt,δ(x)2{x∈D:a<g(x)<b}awhere C is independent of u, a and b.
Remark. Note that the simpliﬁed proofs in [2] and [5, pp.171–182] use the analyticity of harmonic functions ([2, Lemma 1] and [5, Lemma 9.6.2 on p.180]). In general,
L -harmonic functions need not be analytic. Instead of the analyticity, we can use
the unique continuation property of those functions for the present situation.

SEMIBOUNDED PERTURBATION OF GREEN FUNCTION

5

3. Proof of Theorem 1
Proof of Theorem 1. Suﬃciency: Let us recall the resolvent equation∫(3)
G(x, y) = GV (x, y) +GV (x, z)V (z)G(z, y)dz for x, y ∈ D.
DIn view of [14, §4] and [13, Theorem 1.5], it is suﬃcient to show that for any ε > 0
there is a compact subset K of D such that∫
(4)g(z)V (z)G(z, y)dz ≤ εg(y) for y ∈ D,D\K

where we recall g(y) = G(x0 , y).
Suppose (1) holds. Let ϕ(t) = t2/α−1 v(t1/α ), in other words, v(r) = rα−2 ϕ(rα )with α > 1 in (2). By assumption tϕ(t) is nondecreasing. Let b > 0 be suﬃciently
small such that
V (z) = v(δ(z)) = δ(z)α−2 ϕ(δ(z)α ) ≤ Cg(z)ϕ(g(z))δ(z)−2for g(z) < b,where (2) and the monotonicity of tϕ(t) are used. Hence, Corollary 3 with u =G(·, y) yields∫∫
g(z)2
G(z, y)ϕ(g(z))g(z)V (z)G(z, y)dz ≤ Cdzδ(z)2{z∈D:g(z)<b}{z∈D:g(z)<b}∫ b≤ Cg(y)ϕ(t)dt.
0

By an elementary calculation∫ b∫ b1/αrv(r)dr < ∞.

ϕ(t)dt = α00Hence (4) holds with K = {z ∈ D : g(z) ≥ b} for suﬃciently small b > 0.
Necessity: Suppose V is semibounded perturbation of LV . Then by the resolvent
equation (3)∫(5)g(z)V (z)G(z, y)dz ≤ Cg(y) for y ∈ D.
DLet ψ(t) = t2/β−1 v(t1/β ), in other words, v(r) = rβ−2 ψ(rβ ) with 0 < β < 1 in (2).
By assumption tψ(t) is nondecreasing. We choose b > 0 suﬃciently small such thatV (z) = v(δ(z)) = δ(z)β−2 ψ(δ(z)β ) ≥ Cg(z)ψ(g(z))δ(z)−2for g(z) < b,

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HIROAKI AIKAWA

where (2) and the monotonicity of tψ(t) are used. Let 0 < a < b and take y ∈ Dclose to the boundary such that g(y) < a. Then G(·, y) is L -harmonic on {z ∈ D :g(z) > a}. Hence, Corollary 2 with G(·, y) and ψ(t) in place of s and ϕ(t) yields∫∫{z∈D:a<g(z)<b}g(z)V (z)G(z, y)dz ≥ CG(z, y)ψ(g(z)){z∈D:a<g(z)<b}∫ b≥ Cg(y)g(z)2dz
δ(z)2

ψ(t)dt.
aIn view of (5) we have∫ bψ(t)dt ≤ C,awhere C is independent of a and hence∫ b∞>∫ b1/βrv(r)dr.

ψ(t)dt = β00Thus (1) holds. The proof is complete.
References[1] H. Aikawa, Integrability of superharmonic functions and subharmonic functions, Proc. Amer.
Math. Soc. 120 (1994), 109–117.
[2] H. Aikawa, Densities with the mean value property for harmonic functions in a Lipschitz
domain, Proc. Amer. Math. Soc. 125 (1997), 229–234.
[3] H. Aikawa, Norm estimate for the Green operator with applications, Proceedings of Complex
Analysis and Diﬀerential Equations, Marcus Wallenberg Symposium, in honor of Matts Essénat Uppsala in June 1997 (to appear).
[4] H. Aikawa, Norm estimate of Green operator, perturbation of Green function and integrabilityof superharmonic functions, preprint.
[5] H. Aikawa and M. Essén, Potential Theory: Selected Topics, Lecture Notes in Math., vol. 1633,Springer, 1996.
[6] H. Aikawa and M. Murata, Generalized Cranston-McConnell inequalities and Martin boundaries of unbounded domains, J. Analyse Math. 69 (1996), 137–152.
[7] A. Ancona, On strong barriers and an inequality of Hardy for domains in Rn , J. London
Math. Soc. (2) 34 (1986), 274–290.
[8] A. Ancona, First eigenvalues and comparison of Green’s functions for elliptic operators on
manifolds or domains, J. Analyse Math. (to appear).
[9] R. Bañuelos, Intrinsic ultracontractivity and eigenfunction estimates for Schrödinger operators, J. Funct. Anal. 100 (1991), 181–206.
[10] D. S. Jerison and C. E. Kenig, Boundary behavior of harmonic functions in non-tangentiallyaccessible domains, Adv. in Math. 46 (1982), 80–147.
[11] V. G. Maz’ya, Sobolev Spaces, Springer, 1985.
[12] C. Miranda, Partial Diﬀerential Equations of Elliptic Type, Springer, 1970.

SEMIBOUNDED PERTURBATION OF GREEN FUNCTION

7[13] M. Murata, Semismall perturbations in the Martin theory for elliptic equations, Israel J.
Math. (to appear).
[14] N. Suzuki, Martin boundary for ∆ − P , Hiroshima Math. J. 14 (1984), 67–74.
Department of Mathematics, Shimane University, Matsue 690-8504, Japan
E-mail address: haikawa@riko.shimane-u.ac.jp