Default functions and Liouville type theorems based

on symmetric diffusions.

Atsushi Atsuji

Abstract

Default functions appear when one discusses conditions which ensure that a local

martingale is a true martingale. We show vanishing of default functions of Dirichlet

processes enables us to obtain Liouville type theorems for subharmonic functions

and holomorphic maps.

Default functions were introduced in [7] and it is known that vanishing of the default

function of a local martingale implies that it is a true martingale. Positivity of the de-

fault function then indicates the singularity of local martingale. Such local martingales

are called strictly local martingales. Recently strictly local martingales are playing im-

portant roles in the theory of financial bubbles (cf. [19]), so the notion of default function

becomes important in mathematical finance area. We consider them in a different context.

In mathematical analysis of subharmonic functions it is classical and natural to consider

the functions along Brownian motions. A stochastic process derived from a subharmonic

function composed with Brownian motion is a local submartingale. Then if we know that

the process is a true submartingale, which follows from vanishing of the default function

of the submartingale, it effects simpleness and clearness in analysis of subharmonic func-

tions. In this paper we intend to show that this probabilistic notion plays effective roles in

some analysis such as L1-Liouville type theorems of subharmonic functions and Liouville

type theorems for functions satisfying some nonlinear differential inequalities. It covers

and extends the precedent results about L1-Liouville theorem for subharmonic functions

2000 Mathematics Subject Classification. Primary 31C05; Secondary 58J65.

Key words and phrases. Brownian motion on manifolds, strictly local submartingale, subharmonic

function, L1-Liouville theorem, Liouville theorem for holomorphic maps.

Partially supported by the Grant-in-Aid for Scientific Research 17K18741, Japan Society for the

Promotion of Science.

1

on Riemannian manifolds due to P. Li ([13]), Liouville type theorems for functions satis-

fying Yamabe type inequality due to Pigola, Rigoli and Setti ([18]) and Takegoshi ([26]),

and Liouville type theorems for holomorphic maps due to P. Li and S.T. Yau ([15]). We

considered similar problems in [4] to introduce the notion: submartingale property of

subharmonic functions on Riemannian manifold. We treat there Brownian motions and

usual subharmonic functions with respect to Laplace-Beltrami operator and mainly dis-

cussed L1-Liouville theorems. We extend the situation to general symmetric diffusion case

and then we treat general E-subharmonic functions associated with Dirichlet form (E ,F).As an application we consider other Liouville type theorems for functions called strongly

subharmonic functions, which are closely related to Yamabe type differential inequalities.

In this note we consider the following situation. LetM be a separable, locally compact

metric space which supports a diffusion process (Xt, Px) in the following sense. Let (E ,F)be a strongly local, irreducible regular Dirichlet form on L2(m) which defines (Xt, Px)

where m is a Radon measure whose support is M . A typical example in our mind is the

case that M is a complete Riemannian manifold and (E ,F) is defined by

E(u, v) = 12

∫M

⟨∇u,∇v⟩dv (u, v ∈ C∞o (M))

and F = H10 (M) whereH10 (M) is the completion of C∞o (M) by the norm ||u||H10 (M) definedby

||u||2H10 (M) = E1(u, u) := E(u, u) + ||u||2L2(dv),

where dm = dv is a Riemannian volume measure. The corresponding diffusion process is

Brownian motion whose generator is a half of the Laplace -Beltrami operator ∆M defined

from the Riemannian metric.

We define the notion of E-subharmonic functions associated with Dirichlet form (E ,F)as in section 1. It is well-known that u(Xt) is a local submartingale if u is an E-subharmonic function. But it is not always a true submartingale as we saw in [4] when the

diffusion process is a Brownian motion on Riemannian manifolds. In [4] we considered

the conditions on the functions and manifolds that ensure the local submartingales to

be true submartingales when the process is Brownian motion on Riemannian manifolds.

Those are nothing but the conditions for vanishing of default functions. We will define the

default functions of Dirichlet processes associated with (Xt, Px) in section 1 and consider

conditions for vanishing of these default functions.

From simple arguments we obtain L1-Liouville theorems for subharmonic functions in

section 2 and Liouville type theorems for strongly subharmonic functions and holomorphic

maps in sections 3 and 4.

We remark the difference between this paper and our previous work [4]. In the latter

paper we treat the Riemannian case where we can use Harnack inequality for positive

2

harmonic functions:

h(x) ≤ C(r)h(y) for x, y ∈ B(r),

holds for any positive harmonic function h on B(2r) where B(r) is a geodesic ball of

radius r > 0. The constant C(r) is independent of h and can be written in terms of the

lower bound of Ricci curvature on B(r). This inequality enables us to obtain Li-Schoen

estimate of subharmonic functions ([14]) and then we can check the vanishing of default

functions using the estimate. In this paper we treat general case where we cannot expect

the validity of Harnack inequality as above. We consider a condition on the growth of

the Lp-integral of subharmonic function on B(r) instead of use of Ricci curvature (See

Theorem 14 below). Although this condition looks rather rude, we will show that it works

well for some problems as we will see in the strongly subharmonic case of section 3.

In the last section of this paper we also remark the case of subahrmonic functions on

domains in Euclidean spaces which we did not treat in [4].

Acknowledgement: The author would like to thank the reviewer for all valuable and

helpful comments on the manuscript.

1 Submartingale properties for E-subharmonic func-tions

Let M be a locally compact separable metric space and m a Radon measure whose

support is M . Let (E ,F) be a strongly local irreducible regular Dirichlet form on L2(m)associated with an m-symmetric diffusion process (Xt, Px) (See [8] for the definitions of

strong localness, irreducibility and regularity of Dirichlet forms).

It is known (cf. [8]) that for u ∈ F there exists a unique positive measure µ⟨u⟩ calledthe energy measure of u such that

(1.1)

∫M

f(x)dµ⟨u⟩(x) = 2E(uf, u)− E(u2, f), f ∈ F ∩ C0(M).

We define

µ⟨u,v⟩ =1

2(µ⟨u+v⟩ − µ⟨u⟩ − µ⟨v⟩).

Then

E(u, v) = 12µ⟨u,v⟩(M) (u, v ∈ Fb),

where Fb denotes the collection of bounded functions in F . If µ⟨u,v⟩ is absolutely con-tinuous with respect to m, Γ(u, v) denotes the density dµ⟨u,v⟩/dm called a square field

operator.

3

We recall some basic properties of the energy measures. If u1 and u2 ∈ Fb satisfyu1 = u2 on a compact set K, then strong localness and regularity of the Dirichlet form

imply E(u1, ϕ) = E(u2, ϕ) for any ϕ ∈ F ∩ C0(G) and open subset G of K. Hence

(1.2) µ⟨u1⟩(K) = µ⟨u2⟩(K).

Strongly local property also implies the derivation property:

(1.3) dµ⟨uv,w⟩ = ũdµ⟨v,w⟩ + ṽdµ⟨u,w⟩

for u, v, w ∈ Fb where ũ denotes the quasi continuous version of u. We also use a sort ofCauchy-Schwarz inequality:

Lemma 1. ([8, Lemma 5.6.1])∣∣∣∣∫M

f(x)g(x)dµ⟨u,v⟩

∣∣∣∣ ≤ (∫M

f(x)2dµ⟨u⟩(x)

)1/2(∫M

g(x)2dµ⟨v⟩(x)

)1/2holds for f, g ∈ C0(M) and u, v ∈ Fb.

For an open set G ⊂M we define

FG := {u ∈ F | ũ = 0 q.e. on M \G},

and EG denotes the restriction of E to FG. Then (EG,FG) is a strongly local regularDirichlet form on L2(G;m). We say that u is locally in FG if for any relatively compactopen set A ⊂ G there exists w ∈ FG such that u = w m-a.e. on A. Let (FG)loc denotethe set of all functions locally in FG. We also introduce the notion of E-harmonicity. LetU be an open set of M . We say h is E-harmonic on U if h ∈ (FU)loc and E(h, ϕ) = 0 forall ϕ ∈ F ∩ C0(U) ([8, p. 276]).

We put additional assumptions as follows:

(AC) The transition probability p(t, x, dy) is absolutely continuous with respect to m

for t > 0 and x ∈ M . This condition guarantees that several things which hold for q.e.points hold for all x ∈ M . For instance, this guarantees that Xt has continuous samplepaths Px-a.s. for all x ∈M under the strong localness of (E ,F).

(EXH) There exists a non-negative exhaustion function r(x) ∈ Floc (i.e. B(r) ={x | r(x) < r} is relatively compact for ∀r ≥ 0) such that µ⟨r⟩ is absolutely continuouswith respect to m and Γ(r, r) is bounded.

These assumptions are widely satisfied by familiar diffusion processes like Brownian

motions on Riemannian manifolds as mentioned in the introduction. Another example is

a diffusion process whose generator L is a uniformly elliptic operator of divergent form

on a domain in Rd:

L =∑i,j

∂

∂xi

(aij

∂

∂xj

),

4

where (aij(x)) is a locally bounded symmetric matrix valued function satisfying uniform

ellipticity condition. Clearly this is symmetric on L2(dx) where dx denotes the Lebesgue

measure on Rd. In this case we can take r(x) = |x|: Euclidean distance from the originto x as an exhaustion function satisfying (EXH).

We assume the following condition of the diffusions which is important in our analysis:

(CON) (Xt, Px) is conservative.

Throughout this paper we always assume the three conditions: (AC), (EXH) and (CON).

Definition 2. We say that u is E-subharmonic if u ∈ Floc is localy bounded and E(ϕ, u) ≤0 for ∀ϕ ≥ 0, ϕ ∈ F with compact support. If −u is E-subharmonic, we say that u is E-superharmonic.

Note that, in general, for u ∈ Floc u(Xt) can be decomposed into

(1.4) ũ(Xt)− ũ(x) =M [u]t + A[u]t t ∈ [0,+∞), Px-a.s., q.e. x ∈M,

where ũ denotes a quasi continuous version of u, M[u]t is a local martingale additive

functional and A[u]t is an additive functional locally of zero energy (see [8, Theorem 5.5.1]).

Let µ[u] denote the signed smooth measure (i.e. a difference of two smooth measures)

corresponding to A[u]t by Revuz correspondence provided that A

[u]t is locally of bounded

variation. This means that

E(u, ϕ) = −∫M

ϕdµ[u],

holds for any ϕ ∈ F∩C0(M) (see [8, Theorem 5.5.4]). In particular, if u is E-subharmonic,then A

[u]t is a positive continuous additive functional. µ

[u] is called the Revuz measure of

A[u]t . We note the following characterization of the boundedness of variation of A

[u]t and

its Revuz measure.

Proposition 3. ([8, Corollary 5.5.1]) If u ∈ Floc satisfies

E(u, ϕ) = −∫M

ϕdµ for any ϕ ∈ F ∩ C0(M),

where µ is a difference of two smooth Radon measures, then A[u]t is locally of bounded

variation and µ[u] = µ.

We call a signed measure µ a signed smooth Radon measure if µ is a difference of two

smooth Radon measures.

Let U denote the collection of all non-negative Borel measurable E-subharmonic func-tions u satisfying that Ex[ũ(Xt)]

Remark 4. We used some classes of functions in [4]; U0,U1,U2. In this paper we simplifythe presentation and we mainly use U which was represented by U2 in [4]. For a finelycontinuous u ∈ U , A[u]t is a positive continuous additive functional in the strict sense (see[8, Theorem 5.5.5]).

Definition 5 (Submartingale property of E-subharmonic functions). We say that u ∈ Uhas the submartingale property if ũ(Xt) is a continuous submartingale on [0, T ] under Px

for m-a.e. x and any T > 0.

Remark 6. By (AC), we have

Ex[ũ(Xt)] = Ex[u(Xt)]

holds for t > 0 if u ∈ U and ũ is a quasi continuous version of u. Hence if u hassubmartingale property,

u(x) ≤ Ex[u(Xt)]

holds for t > 0 and m-a.e. x.

We here introduce the default function of a function u by that of Dirichlet process

u(X) as in [4]:

Definition 7. We define the default function of an E-subharmonic function u at time Tby

(1.5) Nx(T, u) := limλ→∞

λPx( sup0≤s≤T

ũ(Xs) > λ),

where ũ denotes a quasi continuous version of u.

We can see the above definition is independent of the choice of quasi continuous m-

versions of u if u ∈ U1 from the next lemma. It is easy to see the following (cf. [6],[7]).

Lemma 8. If u ∈ U ,

Nx(T, u) = u(x)− Ex[u(XT )] + Ex[A[u]T ], m-a.e. x ∈M

holds for any T > 0.

If u ∈ U \ U1, both sides of the above equation may be divergent. From this formulaand Definition 7,

Proposition 9. For u ∈ U , Nx(T, u) < ∞ m-a.e. x if and only if Ex[A[u]T ] < ∞ m-a.e.x. Moreover, if Nx(T, u) = 0 for any T > 0, then u has the submartingale property.

6

The following lemma is also basic.

Lemma 10 ([4], Lemma 5). Assume u ∈ U1. If Nx0(t0, u) = 0 for some x0 ∈ M andt0 > 0, then Nx(t, u) = 0 for all t > 0 and m-a.e. x ∈M .

The proof follows from Markov property in the same manner as Brownian motion case

in [4]. The following Lemma can be shown similarly to the above Lemma using Markov

property.

Lemma 11. Assume u ∈ U . If Nx(t0, u) = 0 for m-a.e. x ∈ M and some t0 > 0, thenu ∈ U1.

Proof. Assume ũ(Xt) has the decomposition of (1.4). By definition of additive functionals

and Markov property,

Ex[A[u]2t0

| Gt0 ] = EXt0 [A[u]t0 ] + A

[u]t0 ,

where Gt is an admissible filtration of X. From Lemma 8, vanishing of Nx(t0, u) impliesthat Ex[A

[u]t0 ] ≤ Ex[ũ(Xt0)] m-a.e. x. Hence

Ex

[EXt0 [A

[u]t0 ]]+ Ex[A

[u]t0 ] ≤ Ex

[EXt0 [ũ(Xt0)]

]+ Ex[A

[u]t0 ].

Since Ex[EXt0 [ũ(Xt0)]

]= Ex[ũ(X2t0)]

Under these setting, we obtain a basic criterion for submartingale property.

Theorem 14. If u ∈ U and

lim infr→∞

1

r2

(log

∫B(r)

uαdm+ logm(B(r))

) 2, then u has the submartingale property.

To prove this, by Lemma 13 we will show that

limr→∞

Ex[ũ(Xτr) : τr < t] = 0 m-a.e. x ∈M

holds for some t > 0.

We first estimate Ex[ũ(Xτr)].

Lemma 15. Let r0 > 0 and η > 0. If u is a non-negative E-subharmonic function, thereexists a positive constant C(r0) depending only on r0, η and the bound of Γ(r, r) in (EXH)

such that ∫B(r0)

Ex[ũ(Xτr)]dm(x) ≤ C(r0)(∫

B(r(η+1))

u(x)2dm(x)

)1/2holds for r > r0.

Proof. We first assume that X is transient. We recall an E-orthogonal decomposition

(1.6) Fe = FB(r) ⊕H

where Fe denotes the extended Dirichlet space (see [8, p.170]) and that the projectionP : Fe → H is given by the hitting distribution: Pv(x) = Ex[ṽ(Xτr)]. Let ϕr(x) :=min{2(1+η)

ηr(r(1 + η) − r(x))+, 1} ∈ FB(r(1+η)) and ur = ϕru. Set hr := Pur. Note that

hr = ur on B(r)c. LetMt = hr(Xt). Since hr is E-harmonic on B(r),Mt∧τr is a continuous

Px-martingale for q.e. x ∈M . It holds that

hr(x)2 ≤ Ex[M2t∧τr ] = Ex[⟨M⟩t∧τr ] q.e. x ∈ B(r).

We can take a positive continuous additive functionalKt whose Revuz measure is 1B(r)dµ⟨hr⟩.

Since Kt = ⟨M⟩t for t < τr, we have hr(x)2 ≤ Ex[Kt] for any t > 0, q.e. x ∈ B(r). Forr > r0 ∫

B(r0)

hr(x)2dm(x) ≤

∫B(r0)

Ex[K1]dm(x) ≤∫M

Ex[K1]dm(x).

By the correspondence between positive continuous additive functionals and smooth mea-

sures (see [8, Theorem 5.1.3]), the last term is bounded above by∫ 10

∫B(r)

ps1(x)dµ⟨hr⟩(x)ds ≤ µ⟨hr⟩(B(r)).

8

Since the orthogonal decomposition (1.6) implies

µ⟨hr⟩(B(r(1 + η))) = 2E(hr, hr) ≤ 2E(ur, ur) ≤ µ⟨ur⟩(B(r(1 + η))),

then with a basic property (1.2) of the energy measures we have

µ⟨hr⟩(B(r)) = µ⟨hr⟩(B(r(1 + η)))− µ⟨hr⟩(B(r(1 + η)) \B(r))= µ⟨hr⟩(B(r(1 + η)))− µ⟨ur⟩(B(r(1 + η)) \B(r))≤ µ⟨ur⟩(B(r(1 + η)))− µ⟨ur⟩(B(r(1 + η)) \B(r))= µ⟨ur⟩(B(r)).

Take an arbitrary non-negative ϕ ∈ F ∩ C0(M) such that suppϕ ⊂ B(r(1 + η/2)). Sinceϕur = ϕu and ϕur, ur ∈ Fe, by E-subharmonicity of u and derivation property (1.3) wehave

0 ≤ −2E(ϕ2ur, ur) = −2∫B(r(1+η/2))

ϕũdµ⟨ϕ,u⟩ −∫B(r(1+η/2))

ϕ2dµ⟨u⟩.

Hence by Lemma 1∫B(r(1+η/2))

ϕ2dµ⟨u⟩ ≤ 2∣∣∣∣∫

B(r(1+η/2))

ϕũdµ⟨ϕ,u⟩

∣∣∣∣≤ 2

(∫B(r(1+η/2))

ϕ2dµ⟨u⟩

)1/2(∫B(r(1+η/2))

ũ2dµ⟨ϕ⟩

)1/2.

Namely, ∫B(r(1+η/2))

ϕ2dµ⟨u⟩ ≤ 4∫B(r(1+η/2))

ũ2dµ⟨ϕ⟩.

We take a smooth non-negative function ψ on [0,∞) such that ψ(t) = 1 (t ≤ r), = 0 (t ≥r(1 + η/2)) and |dψ(t)/dt| ≤ 2/(rη).

Set ϕ(x) = ψ(r(x)). Then dµ⟨ϕ⟩ ≤ 4r2η2Γ(r, r)dm ≤C′

r2dm. This implies

µ⟨ur⟩(B(r)) ≤∫B(r(1+η/2))

ϕ2dµ⟨u⟩ ≤4C ′

r2

∫B(r(1+η/2))

u2dm.

Setting C(r0) = 2√C ′m(B(r0))/r0, we obtain the desired estimate.

If X is recurrent, we consider the part process of X on B(R) and (EB(R),FB(R))which is a transient Dirichlet space for R large enough. We remark that we did not use

the conservativeness condition in the transient case. Thus we can deduce the desired

conclusion from the transient case.

We can use Takeda’s inequality in our setting:

Lemma 16 ([25]). Fix r0 > 0. If r > r0, there exist β1, β2 > 0 such that∫B(r0)

Py(τr < t)dm(y) ≤ β1m(B(r))

r − r0exp

(−β2

(r − r0)2

t

).

9

Proof of Theorem 14. By Lemma 13, it suffices to show that for some t > 0

lim infr→∞

Ex[ũ(Xτr) : τr < t] = 0 m-a.e.x.

Take an arbitrary r0 > 0. For r > r0, by Hölder inequality,∫B(r0)

Ex[ũ(Xτr) : τr < t]dm(x) ≤(∫

B(r0)

Ex[ũ(Xτr)α/2]dm(x)

)2/α(∫B(r0)

Px(τr < t)dm(x)

)1−2/α.

Since uα/2 is E-subharmonic, Lemma 15 implies that∫B(r0)

Ex[ũ(Xτr)α/2]dm(x) is bounded

by const. (∫B(r(1+η))

u(x)αdm(x))1/2. With Takeda’s inequality and Fatou’s lemma we

reach the desired result.

2 L1-Liouville theorem and submartingale property

We first note a basic result in [4] which says that vanishing of default functions of E-subharmonic functions implies L1-Liouville theorem.

Proposition 17. If u is a non-negative, m-integrable E-subharmonic function and uhas the submartingale property, then u is constant m-a.e. Namely vanishing of default

function of u implies L1-Liouville theorem.

We give a proof for convenience of readers. The proof presented here is different from

one in [4].

Proof. We first remark that m-integrability of u implies u ∈ U . Take ϕn ∈ C0(M) ∩ Fwith ϕn(x) ↑ 1 (n ↑ ∞) for all x ∈M and set un = ϕnũ. By m-symmetry of X,∫

M

Ex[un(Xt)]dm(x) =

∫M

un(x)dm(x).

From monotone convergence theorem and submartingale property

Ex[ũ(Xt)] = ũ(x), m-a.e. x.

This implies A[u]t = 0, Px-a.s., where A

[u]t is the zero-energy term in the decomposition

(1.4). Let a ∨ b denote max{a, b}. The same argument is available for ũ ∨ a− a, (a > 0)and this implies

Ex[ũ(Xt) ∨ a] = ũ(x) ∨ a, m-a.e. x.

By Itô-Tanaka formula,

ũ(Xt) ∨ a− ũ(X0) ∨ a =∫ t0

1{ũ(Xs)>a}(Xs)dũ(Xs) +1

2Lat ,

where Lat is the local time of ũ(Xt) at a and then Lat = 0. This implies ũ(Xt) never hits

the level a Px-a.s. unless ũ is constant.

10

Example 18. Let M be {|z| < 1} \ {o} in C. Take σ be a smooth function on M suchthat

σ(z) ∼ t−1(log

1

t

)−1(log log

1

t

)−αwith 1/2 < α < 1

when t = |z| → 0. Define a metric ds2 = σ2|dz|2 on M . Take a non-degenerate conformalmetric g satisfying that

g ∼

ds2 around o,the Poincaré metric near |z| = 1.Then g is complete onM and log volB(r) = O(r). The Brownian motion defined from g is

a time-change of a hyperbolic Brownian motion. We can see this process is conservative

since it has the similar behavior to a hyperbolic Brownian motion near the boundary

{|z| = 1}. Set u(z) := − log(2|z| ∧ 1) = (− log |z|)∨ log 2− log 2 ≥ 0. u is a non-negativesubharmonic function integrable with respect to the volume defined from g. We also note

that − log |z| is a positive harmonic function on M and this does not have martingaleproperty. Namely the default function of − log |z| is not vanishing.

Remark 19. In [4] we gave another example in the case that Brownian motion is recur-

rent.

We obtain the following Liouville type theorem on integrable superharmonic func-

tion in similar way to Proposition 17 from the assumption (CON). This was shown by

Grigor’yan [11] for superharmonic functions on stochastically complete Riemannian man-

ifolds.

Lemma 20. If u is an m-integrable, non-negative E-superharmonic function, then u isconstant m-a.e.

We give our L1-Liouville theorem as follows.

Theorem 21. Suppose u is an E-subharmonic function. Set u+ := max{u, 0}.i) Assume there exists α > 2 and 0 ≤ p < 1 such that

lim infr→∞

1

r2(1−p)log

{m(B(r))

∫B(r)

u+(x)αdm(x)

}

If ∫M

|u(x)|1 + r(x)2

dm(x) 0 (cf. [22]). We say thata non-decreasing function f on [0,∞) is moderately increasing if there exists a constantc > 0 such that f(2x) ≤ cf(x) for any x ≥ 0 and that a non-increasing function f on[0,∞) is moderately decreasing if there exists a constant c′ > 0 such that f(2x) ≥ c′f(x)for any x ≥ 0. “Moderately monotone” means moderately increasing or moderatelydecreasing. We remark that there exists a moderately increasing function which is not

regularly varying.

Theorem 22 ([4], [5]). Suppose Ric ≥ −k(r(x)). Let u be a smooth subharmonic functionon M .

12

i) Assume that k(r) is non-decreasing and there exists 0 ≤ p ≤ 1/2 such thatlim infr→∞

k(r)r−2(1−2p)

for 0 < s < t < R if By(R) ⊂ D.Let u be a subharmonic function as in Theorem 22. It is enough to show the default

function of u is vanishing under the assumptions of the theorem. To do this we will

estimate Mx0(r) = sup∂Bx0 (r) u. We may assume u is non-negative. Let y0 be the point

in ∂Bx0(r/2) such that u(y0) = supx∈∂Bo(r/2) u(x). The Li-Schoen estimate (Lemma 11 in

[4]) implies that if Ric ≥ −(n− 1)K0 on By0(2),

(2.3) supx∈∂By0 (1)

u(x) ≤ c3 exp(c4√K0

) 1Vy0(2)

ρ(r)−1∫By0 (2)

u(x)ρ(r(x))dv(x),

where c1, c2 are non-negative constants independent of y0 and ρ is as in the section 5 of

[4]. We need some lower bound of Vy(2). Set κ(r) := k(r)/(n− 1) (n = dimM). We willgive an estimate of Vy(1) in terms of κ(r) as follows.

Take a minimal geodesics γ(t) joining x0 to y0 such that γ(0) = x0, γ(T ) = y0, T =

r(y0) = r/2. We define a set of values in {γ([0, T ])} similarly as in the proof of Theorem2.5 in [14] by

t0 = 0, t1 = 1 + 2, . . . , ti = 2i∑

j=0

2j − 1− 2i = 3(2i − 1).

We take 2 as β in [14]. Let N denote the largest integer such that tN ≤ T . Let xi =γ(ti), Ri = 2

i. Take si ∈ [ti−1, ti] such that κ(si) = mins∈[ti−1,ti] κ(s) (i = 1, 2, . . . , N). Ifx ∈ Bxi(Ri + 2Ri−1), ti−1 − Ri−1 = 2i − 3 ≤ r(x) ≤ ti + Ri + 2Ri−1 = 5 · 2i − 3 (i ≥ 2).Regular variation property of k(t) as (2.1) implies that there exists c5 > 0 such that

maxx∈Bxi (Ri+2Ri−1)

κ(r(x)) ≤ c5κ(r(y))

for any y ∈ Bxi(Ri + 2Ri−1) and for i ≥ 2. Applying (2.2) to our situation with using2Ri−1 =

23(ti − ti−1) , it follows that there exists c6 > 0 such that

Vxi(Ri + 2Ri−1) ≤Ri + 2Ri−1

Riexp

(c6√κ(si)(ti − ti−1)

)Vxi(Ri).

Hence

Vxi−1(Ri−1) ≤ 2n exp(c6√κ(si)(ti − ti−1)

)Vxi(Ri) ≤ 2n exp

(c6

∫ titi−1

√κ(t)dt

)Vxi(Ri),

for every i ≥ 2. Thus we have

Vx1(2) ≤ 2nN exp(c6

∫ tN3

√κ(t)dt

)VxN (RN).

14

By (2.2) Vx0(1) ≤ Vx1(4) ≤ const.Vx1(2). Take sN+1 ∈ [tN−1, T ] such that κ(sN+1) =mins∈[tN−1,T ] k(s). Similarly we can see

VxN (RN) ≤ Vy0(T − tN +RN)

≤ (T − tN +RN)n exp(c7√κ(sN+1)(T − tN +RN − 1)

)Vy0(1)

≤ 2n(N+2) exp

(c8

∫ TtN−1

√κ(t)dt

)Vy0(1).

Here we used that

T − tN +RN − 1 ≤ 4 · 2N , 3 · 2N−1 ≤ T − tN−1,

and regular variation property (2.1). Remark that 2N ≤ r/6 + 1. Hence there existconstants C1, C2 > 0 such that

Vy0(1) ≥ C1r−n exp

(−C2

∫ r/21

√κ(t)dt

)Vx0(1)

for large r > 0. We also see K0 in (2.3) is bounded by const.κ(r). We obtain

sup∂Bx0 (r/2)

u(x) ≤ C3ρ(r)−1rn exp{C4

(∫ r1

√κ(t)dt+

√κ(r)

)}·∫Bx0 (r)

u(x)ρ(r(x))dV (x),

for some C3, C4 > 0. We also remark that the volume growth condition ensures the

conservativeness of the time-changed process Y by Lemma 19 in [4]. Take ρ(t) = (1+t)−2p

with 0 ≤ p < 1 in case of ii) and p = 1 in case of iii). With Takeda’s inequality (Lemma20 in [4]) and changing variables between

∫ r0

√ρ(t)dt and r, the above estimate leads us

to the desired conclusion. We can carry the same procedure as above and obtain the

same conclusion if k(r) is moderately monotone. As for i), that is the case that k(r) is

non-decreasing but does not necessarily satisfy the moderate monotonicity condition, we

can carry out the similar procedure as above using (2.2) more directly:

Vy0(1) ≥ C5r−n exp(−C6r

√κ(r)

)Vx0(1).

We can also see that Vx0(r) ≤ C7 exp(C8r√k(r)

)for some C7, C8 > 0 from (42). So we

can remove the assumption of volume growth in i).

3 Liouville type theorems for strongly subharmonic

functions

Pigola, Rigoli and Setti ([17]) and Takegoshi ([26]) showed :

15

Theorem 24 (Pigola-Rigoli-Setti, Takegoshi). Let M be a non-compact complete Rie-

mannian manifold and v(r) denote the volume of a geodesic ball of radius r > 0 with

center x0. r(x) := d(x0, x). If there exist u ∈ C2(M), C > 0, a > 0 and δ > 0 such that{u > δ} ̸= ∅ and

(3.1) ∆Mu(x) ≥Cu(x)a+1

(1 + r(x))bon {u > δ}

holds for b ≤ 2, then lim infr→∞ log v(r)r2−b = ∞ (b < 2), lim infr→∞log v(r)log r

= ∞ (b = 2).

Remark 25. Pigola-Rigoli-Setti ([17]) first showed the above in the case of b < 2. After

them Takegoshi showed the case of b ≤ 2 by a different method.

A function satisfying (3.1) is called a strongly subharmonic function (cf. [26]). This

inequality is related to Yamabe type differential inequality :

∆Mu(x) + k(x)u(x) ≥ l(x)u(x)1+a.

Using Theorem 24 the authors of [17] and [26] discussed the properties of the solution of

this inequality. Their works are motivated by the Yamabe’s equation taking the following

form.

Let f : (M, g) → (N, h) be a conformal immersion such that f ∗h = u4/(m−2)g (m ≥ 3),f ∗h = ug (m = 2). Then u satisfies :

cm∆M − sgu+Kf∗hu(m+2)/(m−2) = 0 (m ≥ 3),∆M log u− sg +Kf∗hu = 0 (m = 2),

where cm = 4(m−1)/(m−2), sg and Kf∗h are scalar curvatures of g and f ∗h, respectively.We can extend Theorem 24 to the case of our symmetric diffusion case. We obtain

the following result.

Theorem 26. Let ρ be a non-increasing, positive continuous function on [0,∞) such that∫∞0

√ρ(t)dt = ∞. Set Φ(t) :=

∫ t0

√ρ(s)ds. Assume that there exist a locally bounded quasi

continuous u ∈ Floc, a signed smooth Radon measure µ on M and δ > 0 such that

(3.2) m({u > δ}) > 0,

(3.3) E(u, ϕ) = −∫M

ϕdµ for any ϕ ∈ F ∩ C0(M),

and

(3.4) dµ(x) ≥ Cρ(r(x))ua+1(x)dm(x) on {u > δ}

holds for some a > 0 and C > 0. Then

lim infr→∞

logm(r(x) < r)

Φ(r)2= ∞.

16

Let µ[u] denote the Revuz measure of the zero energy part A[u]t in the decomposition of

ũ(Xt) as in (1.4) when A[u]t is locally of bounded variation. By Proposition 3 the condition

(3.3) implies µ = µ[u].

Remark 27. If X is Brownian motion on a Riemannian manifold M and u is smooth

on M , then dµ[u] = 12∆Mudv. Hence (3.3) and (3.4) give a weak form of the differential

inequality (3.1) and we can easily see that Theorem 26 covers Theorem 24.

Corollary 28. Theorem 24 holds replacing the conclusion in the case of b = 2 by

lim infr→∞logm(r(x) 0. Define uδ(x) := u(x) ∨ δ − δ. We have µ = µ[u] and

dµ[uδ] = 1{ũ>δ}dµ+ lδ,

where lδ is a non-negative measure supported by the set {ũ = δ}.

Proof. The first assertion follows from Proposition 3 and then ũ(Xt) is a continuous semi-

martingale. We remark that uδ ∈ Floc if u ∈ Floc. Then ũδ(Xt) can be decomposed as in(1.4). Let A[uδ] denote the zero energy part of the decomposition. Applying Itô-Tanaka

formula (cf. [20]) to ũδ(Xt), we can see that

A[uδ]t =

∫ t0

1{ũ>δ}(Xs)dA[u]s + Lt,

where Lt is the local time of ũδ(Xt) at the level 0. At the same time Lt is a positive

continuous additive functional of Xt. Then a measure lδ corresponds to Lt by Revuz

correspondence (cf. [8]).

By the time-change argument as in the previous section, it is enough to show the

above theorem when ρ = 1. If u is quasi continuous, the above lemma and (3.4) imply

that

dµ[uδ] = 1{u>δ}dµ+ lδ

≥ Cua+11{u>δ}dm.

Namely, uδ satisfies

dµ[uδ] ≥ Cua+1δ dm

17

on M . Thus we assume u and µ satisfy the following (3.5) instead of (3.4):

(3.5) dµ ≥ Cua+1dm on M.

In particular, u is E-subharmonic on M under the assumption (3.5).Next we have to check vanishing of the default function of u ∈ Floc satisfying (3.5).

Lemma 30. If a nonnegative locally bounded u ∈ Floc and µ satisfy (3.3) and (3.5), thereexists a constant C1 > 0 such that∫

B(r/2)

u(x)2+adm(x) ≤ C1V (r),

where V (r) := m({x | r(x) ≤ r}).

Proof. As before, take a nonnegative smooth function ψ on [0,∞) such that ψ(t) = 1 (0 ≤t ≤ r/2), = 0 (t ≥ r) and |ψ′(t)|, |ψ′(t)|2+a/ψ(t)2 are bounded. Set ϕ(x) := ψ(r(x)).Since Γ(ϕ, ϕ) = |ψ′(r(x))|2Γ(r, r) ≤ const.|ψ′(r(x))|2, |ψ′(r(x))| and |ψ′(r(x))|2+a/ϕ(x)2

are bounded, Γ(ϕ, ϕ)(2+a)/2ϕ−2 is bounded. By derivation property,

−2E(ϕ2u, u) = 2∫B(r)

ϕũdµ⟨ϕ,u⟩ −∫B(r)

ϕ2dµ⟨u⟩

≤ 2(∫

B(r)

ũ2dµ⟨ϕ⟩

)1/2(∫B(r)

ϕ2dµ⟨u⟩

)1/2.

As in the proof of Lemma 15, since u is E-subharmonic,∫B(r)

ϕ2dµ⟨u⟩ ≤ 2(∫

B(r)

u2dµ⟨ϕ⟩

)1/2(∫B(r)

ϕ2dµ⟨u⟩

)1/2.

Hence ∫B(r)

ϕ2dµ⟨u⟩ ≤ 4∫B(r)

u2dµ⟨ϕ⟩,

Then from (3.3), (3.4) and regularity of (E ,F),

C

∫B(r)

ϕ2u2+adm ≤ −E(ϕ2u, u) ≤ 2∫B(r)

u2dµ⟨ϕ⟩.

By Hölder inequality, there exists a constant c′ > 0 such that∫B(r)

u2dµ⟨ϕ⟩ ≤ V (r)1/q(∫

B(r)

Γ(ϕ, ϕ)(2+a)/2ϕ−2ϕ2u2+adm

)2/(2+a)≤ c′V (r)1/q

(∫B(r)

ϕ2u2+adm

)2/(2+a),

where q = 2/a+ 1. Hence we have∫B(r)

ϕ2u2+adm ≤(2c′

C

)qV (r).

18

From this we have the following.

Proposition 31. Assume

lim infr→∞

log V (r)

r2 0}) > 0 andlim infr→∞

log V (r)r2

< ∞. By Proposition 31, we can take R0 = R0(x) > 0 for m-a.e. xon {u > 0} such that Ex[u(XτR) : τR ≤ t] ≤ u(x) for R ≥ R0 and 0 < t < T0. Thedecomposition (1.4) with (3.5) implies that for R > r(x) ∨R0

Ex[u(Xt) : τR > t] ≥ Ex[u(Xt∧τR)]−u(x) ≥ CEx[∫ t∧τR

0

u(Xs)1+ads

]∀t > 0, m−a.e. x.

Fix x0 ∈ M for which the above inequality holds. By Jensen’s inequality and positivityof u, for R ≥ r(x0) ∨R0

Ex0 [u(Xt)1+a1{τR>t}] ≥ C1+aEx0

[∫ t0

u(Xs)1+a1{τR>s}ds

]1+a.

Let fR(t) := Ex0 [∫ t0u(Xs)

1+a1{τR>s}ds] for R ≥ r(x0) ∨ R0. By local boundedness ofu, fR(t) is finite for each t and R. We also remark that fR(t) > 0 (0 < t < T0) for

R large enough since m({u > 0}) > 0 and we assume (AC). Then fR satisfies f ′R(t) ≥C1+afR(t)

1+a for t ∈ (0, T0). This implies fR(s)−a − fR(t)−a ≥ aC1+a(t − s), (T0 >t > s > 0). If T0 = ∞, let t → ∞. If T0 < ∞ and fR(s) → ∞ (R → ∞), letR → ∞. In both cases we reach contradiction. If T0 < ∞ and supR fR(s) < ∞, we haveess. supxEx[

∫ t0u(Xs)

1+ads] ≤ a−1C−(1+a)(T0 − t)−1 for 0 < t < T0. Markov property of Ximplies Ex[

∫ t0u(Xs)

1+ads] < ∞ for ∀t > 0, m-a.e. x. This means u ∈ U . From this andLemma 13, Nx(t, u) = 0 for t > 0,m-a.e x. We obtain f(s)

−a−f(t)−a ≥ aC1+a(t−s) (∀t >s > 0) where f(t) = Ex[

∫ t0u(Xs)

1+ads]. This leads to contradiction.

From Theorem 26 with the same argument as in [26] we have the following result from

which we will obtain some Schwarz type results as in [18].

Corollary 32. Let M and ρ be as in Theorem 26. Assume a nonnegative continuous

u ∈ Floc and a singed smooth Radon measure µ on M satisfy (3.3) and

dµ+ (ku− lua+1)dm ≥ 0,

19

where k, l are continuous functions on M satisfying that there exists a constant H > 0

such that

k(x) ≤ Hl(x), l(x) ≥ ρ(r(x)) for all x ∈M.

If lim infr→∞logm(r(x) H/(1− δ)} ̸= ∅ and this is open. Then on this set,

dµ(x) ≥ δl(x)u(x)1+adm(x) ≥ δρ(x)u(x)1+adm(x).

Theorem 26 implies this is impossible.

Pigola et al. [18] showed that some Schwarz lemma type results using a result like the

above. We can extend their results slightly. For instance, we have the following result

about harmonic maps with some dilatation condition. Let f : M → N be a harmonicmap between Riemannian manifolds (M, gM) and (N, gN). Let λ1 ≥ λ2 ≥ . . . λs ≥ 0 beeigenvalues of f∗. We consider a condition on dilatation as

(3.6) λ1(x) ≤ a(x)λ2(x)

holds on M for a positive continuous function a(x). Assume

RicM ≥ −b(x) (x ∈M), SectN ≤ −K(z) (z ∈ N) and K(f(x)) ≤ −c(x),

where b, c are continuous non-negative functions on M and K(z) is a continuous non-

negative function on N . Let m and n be the dimensions ofM and N , respectively. Pigola

et al.[18] showed that Bochner-Weitzenböck formula implies

∆Me(x) ≥ −2b(x)e(x) +4

c(m,n)

c(x)

a(x)e(x)2,

where e(x) is the energy density of f , namely e(x) = trgMf∗gN , and c(m,n) = min{m,n}.

Hence we have the following. As before r(x) is a distance function from a reference point

to x and v(r) denotes the Riemannian volume of the set {x | r(x) ≤ r}.

Proposition 33. Let ρ be a non-increasing, positive continuous function on [0,∞) suchthat

∫∞0

√ρ(t)dt = ∞ and set Φ(t) :=

∫ t0

√ρ(s)ds. If 4

c(m,n)c(x)a(x)

≥ ρ(r(x)) with∫∞√

ρ(t)dt =

∞ and lim infr→∞ log v(r)Φ(r)2

4 Liouville type theorems for holomorphic maps

We give a remark on a Liouville type theorem of holomorphic maps which is a general-

ization of one obtained by [15] since it is closely related to the argument of the previous

sections.

Let M be a complete Kähler manifold equipped with a Kähler metric gM, N a Hermi-tian manifold and f : M → N a holomorphic map. We here use the following notations:B(r) := {x ∈ M| r(x) < r}, K(y) denotes the holomorphic bisectional curvature of N ,e(x) := trgMf

∗gN denotes the energy density of f .

Let R(x) be a continuous function on M satisfying Ric(ξ, ξ) ≥ R(x) for all ξ ∈ TxMwith ||ξ|| = 1 and R−(x) := max{0,−R(x)}. A fundamental inequality in analysis of e(x)comes from Chern-Lu formula ([16], cf. [12]):

(4.1)1

2∆M log e(x) ≥ −K(f(x))e(x)−R−(x) if e(x) ̸= 0.

As we will see below, we can modify and adjust the method in the previous sections to

this equation, and then we have the following result.

Theorem 34. Assume Brownian motion on M is transient. If K(f(x)) ≤ −ρ(r(x)) forsome ρ satisfying the same condition as Theorem 26,

∫MR−(x)dv(x)

holds weakly where lδ is a measure supported by the set {e(x) = δ}. From (4.2) withρ = 1

1

2∆Mu ≥

1

21{e(x)>δ}δe

u(x) − g(x) ≥ δ4u(x)2 − g(x).

We wish to obtain a general version of the above theorem so that it holds under the same

situation as the previous sections 1, 2 and 3. Let M be a general state space as in section

1 and we assume the same assumptions on the state space M and (Xt, Px) as section

1. Then we rewrite (4.2) generally to (4.3) below and we show a perturbed version of

Theorem 26 as follows. For an m-integrable function g set

Rg(x) := Ex

[∫ ∞0

g(Xs)ds

].

Theorem 37. Suppose that (Xt, Px) is transient. Let u ∈ Floc be nonnegative locallybounded quasi continuous, µ a signed smooth Radon measure satisfying (3.3) and g a

nonnegative m-integrable function such that Rg ∈ Floc. If

(4.3) dµ ≥ C(u(x)a+1 − g(x)

)dm(x) on M

holds for some a > 0 and C > 0, then

lim infr→∞

logm (B(r))

r2= ∞,

where B(r) = {x ∈M | r(x) < r}.

Transience enables us to reduce the problem to Theorem 26 similarly to [18].

Lemma 38. Let a > 0, g ∈ L1(m) and u ∈ Floc be as in Theorem 37. Let

w(x) = (u(x)−Rg(x)) ∨ 0.

Then µ[w] exists and satisfies

dµ[w] ≥ Cwa+1dm on M,

where C > 0 is the constant appearing in (4.3).

Proof. First note that transience implies Ex[∫∞0g(Xs)ds] < ∞ and that w is quasi con-

tinuous and 0 ≤ w(x) ≤ u(x). We also have dµ[Rg] = −gdm. As we saw in Lemma29,

dµ[w] = 1{u>Rg}dµ[u] − 1{u>Rg}dµ[Rg] + dl0,

where l0 is a measure supported by the set {u = Rg}. Then

dµ[w] ≥ 1{u>Rg}(dµ[u] + gdm) ≥ C1{u>Rg}ua+1dm ≥ Cwa+1dm.

22

Proof of Theorem 37 and Theorem 34. Theorem 37 immediately follows from the above

lemma and Theorem 26. Let (E ,F) be associated with Brownian motion on M whosegenerator is 1

2∆M. We set u(x) = (log e(x)) ∨ (log δ) − log δ ≥ 0 for δ > 0 and g(x) =

R−(x). Since R− is continuous and integrable on M, Rg ∈ C2(M) and then Rg ∈ Floc.If e(x) is not constant, there exists δ > 0 such that the assumptions in Theorem 37 holds

with a = 1 and µ = µ[u] as remarked before. Hence the volume growth condition in

Theorem 34 is impossible. If e(x) is a non-zero constant, (4.2) implies R− is bounded

below by a positive constant onM. Since R−(x) is integrable onM, M has finite volume.This contradicts transience of Brownian motion on M. Therefore e(x) = 0 for all x ∈ Munder the setting of Theorem 34. This implies f is constant.

5 Subharmonic functions on Euclidean domains

As mentioned in the introduction, we discuss similar problems to the previous sections

for continuous subharmonic functions on domains in Euclidean space Rd. We can apply

the method in the previous sections to the following situation.

We assume that D is a regular bounded domain throughout this section. Let δD(x)

denote the distance from x ∈ D to the boundary ∂D of D. Define a quasi-hyperbolicdistance kD on D by

kD(x, y) = infγ

∫γ

ds

δD,

where inf is taken over all curves joining x and y.

gD(x, y) denotes the Green function of Euclidean Laplacian ∆ with Dirichlet boundary

condition on D. Define a Radon measure m on D by dm = 1δD(x)2

dx where dx denotes

the Lebesgue measure on Rd.

Under this situation we have the following Harnack inequality which is important in

the analysis on D.

Proposition 39. (cf. [1]) There exist constants C1, C2 > 0 depending only d such that if

h is a positive harmonic function,

|∇h|(x)h(x)

≤ C11

δD(x),

and

exp (−C2(kD(x, y) + 1)) ≤h(x)

h(y)≤ exp (C2(kD(x, y) + 1)) .

We define a diffusion process (Xt, Px) by a time-changed process of Brownian motion:

XAt = Bt,

23

where Bt is the Euclidean Brownian motion whose generator is12∆ and

At =

∫ t0

1

δD(Bs)2ds.

We remark that the generator L of (Xt, Px) is12δD(x)

2∆ which is symmetric on L2(m)

and the corresponding Dirichlet form is E(u, v) = 12

∫D⟨∇u,∇u⟩dx (u, v ∈ C∞0 (D)) on

L2(m).

Fix x0 ∈ D and set gD(x) = gD(x0, x). Let r(x) = − log gD(x) and B(r) = {x ∈D | r(x) < r}.

Lemma 40. r(x) is an exhaustion function satisfying the condition (EXH) in section 1.

Proof. By direct computation, we have Γ(r, r) = |∇gD|2

g2D. By Proposition 39, we have

|∇gD|2g2D

≤ const. 1δ2D

near the boundary of D. This shows (EXH).

We also see X satisfies (CON).

Proposition 41. (Xt, Px) is conservative.

Proof. By Itô’s formula and Dambis-Dubins-Schwarz theorem (cf. [20])

r(Bt)− r(B0) = b(Gt) +1

2Gt,

where b(t) is the standard Brownian motion on R and

Gt =

∫ t0

|∇gD|2

g2D(Bs)ds.

By Proposition 39, Gt ≤ const.At (t > 0). On the other hand, since D is regular,limt→σD gD(Bt) = 0 a.s. if B0 ̸= x0 where σD = inf{t > 0 : Bt /∈ D}. Hencelimt→σD r(Bt) = +∞ a.s. Then By the law of iterated logarithm of Brownian motion(cf. [20])

lim supt→∞

|bt|√2t log log t

= 1, a.s.

This implies limt→σD At = ∞ a.s. This property holds independently of the choice of B0.Hence the life time ζ of X satisfies ζ = AσD = ∞ Px-a.s. for any x ∈ D.

We introduce a characteristic of the domain like a diameter by the quasi-hyperbolic

distance:

κ(r) = supx∈B(r)

kD(x0, x).

It is known (cf. [9]) that the quasi-hyperbolic distance kD satisfies

(5.1) kD(x, x0) ≥ log1

δD(x)+ log δD(x0)

if δD(x) ≤ δD(x0). We give an estimate of m(B(r)) in terms of κ(r). The next lemmaimmediately follows from (5.1).

24

Lemma 42.

(5.2) m(B(r)) ≤ δD(x0)−2e2κ(r)vol(B(r)),

where vol(B) is the Euclidean volume of B.

From Proposition 39 and the method of the previous sections we have the following

estimate analogous to Theorem 1.2 in [14].

Lemma 43. Let η > 0. There exist constants C3, C4 > 0 such that for any positive

continuous subharmonic function u on D and r > r0,

supx∈B(r)

u(x) ≤ C3eC4κ(r)(∫

B(r(1+η))

u(x)2dm(x)

)1/2.

Proof. Fix r0 with r0 < r. Since u is subharmonic and u(Xτr(1+η/2)∧t) is a submartingale,

by Proposition 39 and Lemma 15 we have

supx∈B(r)

u(x) ≤ supx∈B(r)

Ex[u(Xτr(1+η/2))]

≤ C(r) infx∈B(r0)

Ex[u(Xτr(1+η/2))]

≤ C(r)m(B(r0))

∫B(r0)

Ex[u(Xτr(1+η/2))]dm(x)

≤ C(r)C(r0)m(B(r0))

(∫B(r(1+η))

u(x)2dm(x)

)1/2,

where C(r) = exp {C2(κ(r) + 1)} and C(r0) is the constant in Lemma 15.

By applying the same argument of [14] to Lemma 43 we have the following.

Lemma 44. Let η > 0 and 0 < p ≤ 2. There exist constants C5, C6 > 0 such that forany positive continuous subharmonic function u on D and r > r0,

supx∈B(r)

u(x)p ≤ C5eC6κ(r)∫B(r(1+η))

u(x)pdm(x).

We have dealt with general domains as D, but it seems relevant to treat some wide

class of domains where our method works well. John domains are suitable for this aim.

We can give some good estimates on κ(r) and m(B(r)) there.

Lemma 45. If D is a bounded John domain, then κ(r) = O(r) and logm(B(r)) = O(r).

25

Proof. It is known that if D is a John domain, then D satisfies the capacity density

condition (CDC). It is also known ([3]) that CDC implies

gD(x) ≤ const.δD(x)β,

for some β > 0. From this m(B(r)) ≤∫B(r)

dxδD(x)2

≤ const. exp(2r/β) since D is bounded.Also John domains satisfy the following quasi-hyperbolic metric condition (cf. [10, Lemma

3.11]):

kD(x0, x) ≤ c log1

δD(x)+ c′,

for some c, c′ > 0. Hence κ(r) = O(r).

Proposition 46. Suppose D is a John domain and ρ is a non-decreasing function on

(0,∞) satisfying

lim inft→∞

log ρ(t)

t2

This also implies u ∈ U with respect to Y . From Proposition 46 with (5.3) we see thesubmartingale property of u with respect to Y . It implies that (5.4) is possible only if

u = 0.

Remark 48. Aikawa [2] gave some results similar to the above by simpler methods using

co-area formula.

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Keio University

3-14-1, Hiyoshi

Yokohama, 223-8522

Japan

E-mail address: atsuji@math.keio.ac.jp

29