Volume 8 (2007), Issue 2, Article 36, 24 pp.
DIRICHLET GREEN FUNCTIONS FOR PARABOLIC OPERATORS WITH
SINGULAR LOWER-ORDER TERMS
LOTFI RIAHI
D EPARTMENT OF M ATHEMATICS ,
NATIONAL I NSTITUTE OF A PPLIED S CIENCES AND T ECHNOLOGY,
C HARGUIA 1, 1080, T UNIS , T UNISIA
Lotfi.Riahi@fst.rnu.tn
Received 15 March, 2006; accepted 10 April, 2007
Communicated by S.S. Dragomir
A BSTRACT. We prove the existence and uniqueness of a continuous Green function for the parabolic operator L = ∂/∂t − div(A(x, t)∇x ) + ν · ∇x + µ with the initial Dirichlet boundary
condition on a C 1,1 -cylindrical domain Ω ⊂ Rn × R, n ≥ 1, satisfying lower and upper estimates, where ν = (ν1 , . . . , νn ), νi and µ are in general classes of signed Radon measures
covering the well known parabolic Kato classes.
Key words and phrases: Green function, Parabolic operator, Initial-Dirichlet problem, Boundary behavior, Singular potential,
Singular drift term, Radon measure, Schrödinger heat kernel, Parabolic Kato class.
2000 Mathematics Subject Classification. 34B27, 35K10.
1. I NTRODUCTION
In this paper we are interested in the parabolic operator
L = L0 + ν · ∇x + µ,
where L0 = ∂/∂t − div(A(x, t)∇x ) on Ω = D×]0, T [, D is a bounded C 1,1 -domain in Rn , n ≥
1 and 0 < T < ∞. The matrix A is assumed to be real, symmetric, uniformly elliptic with
Lipschitz continuous coefficients, ν = (ν1 , . . . , νn ), νi and µ are signed Radon measures on
Ω. Recall that Zhang studied the perturbations L0 + B(x, t) · ∇x [37, 40] and L0 + V (x, t)
[38, 39] of L0 with B and V in some parabolic Kato classes. Using the well known results
by Aronson [1] for parabolic operators with coefficients in Lp,q -spaces and an approximation
argument, he proved, in both cases, the existence and uniqueness of a Green function G for the
initial-Dirichlet problem on Ω. The existence of the Green function allowed him to solve some
initial boundary value problems. In [28] and [31], we have established two-sided pointwise
estimates for the Green functions describing, completely, their behavior near the boundary.
These estimates are used to prove some potential-theoretic results, namely, the equivalence of
I want to sincerely thank the referee for his/her interesting comments and remarks on a earlier version of this paper. I also want to sincerely
thank Professor El-Mâati Ouhabaz for some interesting remarks on the last section, and Professor Minoru Murata for interesting discussions
and comments about the subject when I visited Tokyo Institute of Technology, and I gratefully acknowledge the financial support and hospitality
of this institute.
075-06
2
L OTFI R IAHI
harmonic measures [31], the coincidence of the Martin boundary and the parabolic boundary
[27]; and they simplify proofs of certain known results such as the Harnack inequality, the
boundary Harnack principles [28], etc. In the elliptic setting, similar estimates are well known
(see [3, 8, 11, 12, 43]) and have played a major role in potential analysis; for instance they were
used to prove the well known 3G-Theorems and the comparability of perturbed Green functions
(see [10, 13, 26, 29, 30, 32, 43]).
Our aim in this paper is to introduce general conditions on the measures ν and µ which guarantee the existence and uniqueness of a continuous L-Green function G for the initial-Dirichlet
problem on Ω satisfying two-sided estimates like the ones in the unperturbed case. In fact, we
establish the existence of G when ν and µ are in general classes covering the parabolic Kato
classes used by Zhang [37] – [40]. Some partial counterpart results in the elliptic setting have
recently been proved in [13, 30] and are based on new 3G-Theorems which cover the classical
ones due to Chung and Zhao [3], Cranston and Zhao [4] and Zhao [43]. In the parabolic setting it is not clear whether versions of these theorems hold. Here we establish basic inequalities
(Lemmas 3.1 – 3.3 below) which imply the elliptic new 3G-Theorems for all dimensions n ≥ 1,
and which are a key in proving the existence result. The paper is organized as follows.
In Section 2, we give some notations and state some known results. In Section 3, we prove
some useful inequalities that will be used in the next sections. Parabolic versions of the elliptic
3G-Theorems [13, 26, 29, 30, 32] are proved. In Section 4, we introduce general classes of
drift terms ν and potentials µ denoted by Kcloc (Ω) and Pcloc (Ω), respectively, and we study some
of their properties. In Section 5, we prove the existence and uniqueness of a continuous LGreen function G for the initial-Dirichlet problem on Ω satisfying lower and upper estimates
as in the unperturbed case, when ν and µ are in the classes Kcloc (Ω) and Pcloc (Ω), with small
norms M c (ν) and N c (µ− ), respectively (see Theorem 5.6 and Corollary 5.7). In particular,
these results extend the ones proved in [14, 28, 31, 37, 38] to a more general class of parabolic
operators. In Section 6, we consider the time-independent case A = A(x), ν = 0, µ = V (x)dx
and we establish global-time estimates for Schrödinger heat kernels.
Throughout the paper the letters C, C 0 . . . denote positive constants which may vary in value
from line to line.
2. N OTATIONS AND K NOWN R ESULTS
We consider the parabolic operator
∂
− div(A(x, t)∇x ) + ν · ∇x + µ
∂t
on Ω = D×]0, T [, where D is a C 1,1 -bounded domain in Rn , n ≥ 1 and 0 < T < ∞. By
a domain we mean an open connected set. For n = 1, D =]a, b[ with a, b ∈ R, a < b. We
assume that the matrix A is real, symmetric, uniformly elliptic, i.e. there is λ ≥ 1 such that
λ−1 kξk2 ≤ hA(x, t)ξ, ξi ≤ λkξk2 , for all (x, t) ∈ Ω and all ξ ∈ Rn with λ-Lipschitz continuous
coefficients on Ω, ν = (ν1 , . . . , νn ), νi and µ are signed Radon measures. When ν = 0 and
µ = 0, we denote L by L0 . We denote by G0 the L0 -Green function for the initial-Dirichlet
problem on Ω. In the time-independent case, we denote by g0 (resp. g−∆ ) the Green function of
L0 = − div(A(x)∇x ) (resp. −∆) with the Dirichlet boundary condition on D. By [12], there
exists a constant C = C(n, λ, D) > 0 such that C −1 g−∆ ≤ g0 ≤ Cg−∆ . Using this comparison
and the estimates on g−∆ proved in [8, 11, 43] for n ≥ 3, in [3] for n = 2 and the formula
L=
g−∆ (x, y) =
(b − x ∨ y)(x ∧ y − a)
b−a
for
n = 1,
we have the following.
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D IRICHLET G REEN F UNCTIONS FOR PARABOLIC O PERATORS
3
Theorem 2.1. There exists a constant C = C(n, λ, D) > 0 such that, for all x, y ∈ D,
C −1 Ψ(x, y) ≤ g0 (x, y) ≤ CΨ(x, y),
where
Ψ(x, y) =













d(x)d(y)|x−y|2−n
d(x)d(y)+|x−y|2
Log 1 +
d(x)d(y)
|x−y|2
if n ≥ 3;
d(x)d(y)
|x−y|+
√
if n = 2;
if n = 1,
d(x)d(y)
with d(x) = d(x, ∂D), the distance from x to the boundary of D.
For a > 0, x, y ∈ D and s < t, let
1
|x − y|2
Γa (x, t; y, s) =
exp −a
,
(t − s)n/2
t−s
d(x)
d(y)
γa (x, t; y, s) = min 1, √
min 1, √
Γa (x, t; y, s),
t−s
t−s
and
d(y)
Γa (x, t; y, s)
√
ψa (x, t; y, s) =
= min 1, √
.
t−s
t−s
The following estimates on the L0 -Green function G0 were recently proved in [31].
ψa∗ (y, t; x, s)
Theorem 2.2. There exist constants k0 , c1 , c2 > 0 depending only on n, λ, D and T such that
for all x, y ∈ D and 0 ≤ s < t ≤ T ,
(i) k0−1 γc2 (x, t; y, s) ≤ G0 (x, t; y, s) ≤ k0 γc1 (x, t; y, s),
(ii) |∇x G0 |(x, t; y, s) ≤ k0 ψc1 (x, t; y, s) and
(iii) |∇y G0 |(x, t; y, s) ≤ k0 ψc∗1 (x, t; y, s).
3. BASIC I NEQUALITIES
In this section we prove some basic inequalities which are a key in obtaining the existence
results.
Lemma 3.1 (3γ-Inequality). Let 0 < a < b. Then for any 0 < c < min(a, b − a), there exists a
constant C0 = C0 (a, b, c) > 0 such that, for all x, y, z ∈ D, s < τ < t,
γa (x, t; z, τ )γb (z, τ ; y, s)
d(z)
d(z)
≤ C0
γc (x, t; z, τ ) +
γc (z, τ ; y, s) .
γa (x, t; y, s)
d(x)
d(y)
Proof. We may assume s = 0. Let x, y, z ∈ D, 0 < τ < t. We have
(3.1)
γa (x, t; z, τ )γb (z, τ ; y, 0) = wΓa (x, t; z, τ )Γb (z, τ ; y, 0),
where
d(x)
w = min 1, √
t−τ
d(z)
min 1, √
t−τ
d(z)
min 1, √
τ
d(y)
min 1, √
τ
.
Let ρ ∈]0, 1[ which will be fixed later.
Case 1. τ ∈]0, ρt]. We have
1
1
≤
.
n/2
(t − τ )
((1 − ρ)t)n/2
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4
L OTFI R IAHI
Combining with the inequality
|x − z|2 |z − y|2
|x − y|2
+
≥
,
t−τ
τ
t
for all τ ∈]0, t[,
we obtain
Γa (x, t; z, τ )Γb (z, τ ; y, 0) ≤
(3.2)
1
Γb−a (z, τ ; y, 0)Γa (x, t; y, 0).
(1 − ρ)n/2
Moreover, using the inequalities
αβ
αβ
≤ min(α, β) ≤ 2
,
α+β
α+β
for α, β > 0, and |d(z) − d(y)| ≤ |z − y|, we have
d(z)
d(y)
d(z)
|z − y|
min 1, √
≤2
min 1, √
1+ √
d(y)
t−τ
t−τ
t−τ
2 d(z)
d(y)
|z − y|
≤
min 1, √
1+ √
(3.3)
1 − ρ d(y)
τ
t
Combining (3.1) – (3.3), we obtain, for all τ ∈]0, ρt],
γa (x, t; z, τ )γb (z, τ ; y, 0) ≤
2
(1 − ρ)
n+3
2
d(z)
γc (z, τ ; y, 0)γa (x, t; y, 0)
d(y)
|z − y|
|z − y|2
× 1+ √
exp −(b − a − c)
.
τ
τ
Using the inequality (1 + θ) exp(−αθ2 ) ≤ 1 + α−1/2 , for all α, θ ≥ 0, it follows that
γa (x, t; z, τ )γb (z, τ ; y, 0) ≤ C0
(3.4)
d(z)
γc (z, τ ; y, 0)γa (x, t; y, 0),
d(y)
where C0 = C0 (a, b, c, ρ) > 0.
Case 2. τ ∈ [ρt, t[. If |z − y| ≥ ( ab )1/2 |x − y|, then
|z − y|2
|x − y|2
(3.5)
exp −b
≤ exp −a
.
τ
t
If |z − y| ≤ ( ab )1/2 |x − y|, then
|x − z| ≥ |x − y| − |z − y| ≥
1−
a 12 b
|x − y|,
which yields
|x − z|2
exp −a
t−τ
!
a 12 2
a + c |x − z|2
a − c |x − y|2
≤ exp −
exp −
1−
2
t−τ
2
t−τ
b
!
a 12 2
a + c |x − z|2
a − c |x − y|2
≤ exp −
exp −
1−
.
2
t−τ
2
(1 − ρ)t
b
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D IRICHLET G REEN F UNCTIONS FOR PARABOLIC O PERATORS
5
Now taking ρ so that
(a − c) 1 −
a
b
12 2
2a(1 − ρ)
= 1,
we obtain
a + c |x − z|2
|x − y|2
|x − z|2
≤ exp −
exp −a
.
(3.6)
exp −a
t−τ
2
t−τ
t
From (3.5) and (3.6), we have
1
(3.7)
Γa (x, t; z, τ )Γb (z, τ ; y, 0) ≤ n/2 Γ a+c (x, t; z, τ )Γa (x, t; y, 0).
2
ρ
Note that (3.7) is similar to the inequality (3.2). Then by the same method used to prove (3.4),
we obtain
d(z)
(3.8)
γa (x, t; z, τ )γb (z, τ ; y, 0) ≤ C0
γc (x, t; z, τ )γa (x, t; y, 0).
d(x)
Combining (3.4), (3.8) and using the fact that
1 2
(a − c) 1 − ab 2
= 1,
2a(1 − ρ)
we get the inequality of Lemma 3.1 with C0 = C0 (a, b, c) > 0.
Lemma 3.2. Let 0 < a < b. Then for any 0 < c < min(a, b − a), there exists a constant
C1 = C1 (a, b, c) > 0 such that, for all x, y, z ∈ D, s < τ < t,
γa (x, t; z, τ )ψb (z, τ ; y, s)
≤ C1 [ψc (x, t; z, τ ) + ψc∗ (z, τ ; y, s)] .
γa (x, t; y, s)
Proof. We may assume that s = 0. Letting x, y, z ∈ D, 0 < τ < t, we have
(3.9)
where
γa (x, t; z, τ )ψb (z, τ ; y, 0) = wΓa (x, t; z, τ )Γb (z, τ ; y, 0),
d(x)
w = min 1, √
t−τ
Let ρ ∈]0, 1[ that will be fixed later.
d(z)
min 1, √
t−τ
d(y)
min 1, √
τ
1
√ .
τ
Case 1. τ ∈]0, ρt]. As in (3.2), we have
1
Γb−a (z, τ ; y, 0)Γa (x, t; y, 0)
(1 − ρ)n/2
1
(3.10)
≤
Γc (z, τ ; y, 0)Γa (x, t; y, 0)
(1 − ρ)n/2
Moreover, by using the same inequalities as in (3.3), we obtain
2
d(x)
d(y)
|z − y|
1
4
d(z)
√
√
√
√
√
1
+
.
(3.11) w ≤
min
1,
min
1,
min
1,
(1 − ρ)3/2
τ
τ
τ
t
t
Combining (3.9) – (3.11) and using the inequality
2
1
2
2
(1 + θ) exp(−αθ ) ≤ 2 1 + √
,
α
for all α, θ ≥ 0, it follows that
Γa (x, t; z, τ )Γb (z, τ ; y, 0) ≤
γa (x, t; z, τ )ψb (z, τ ; y, 0) ≤ C1 ψc∗ (z, τ ; y, 0)γa (x, t; y, 0),
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6
L OTFI R IAHI
with
1
C1 = 8 1 + √
b−a−c
(1 − ρ)−
n+3
2
.
Case 2. τ ∈ [ρt, t[. If |z − y| ≥ ( ab )1/2 |x − y|, then
|z − y|2
|x − y|2
(3.12)
exp −b
≤ exp −a
.
τ
t
If |z − y| ≤ ( ab )1/2 |x − y|, then |x − z| ≥ (1 − ( ab )1/2 )|x − y|, which yields
!
a 1/2 2
|x − z|2
|x − z|2
|x − y|2
exp −a
≤ exp −c
exp −(a − c)
1−
.
t−τ
t−τ
(1 − ρ)t
b
Now taking ρ so that
(a − c) 1 −
2
a 1/2
b
a(1 − ρ)
= 1,
we obtain
|x − z|2
exp −a
t−τ
(3.13)
|x − z|2
≤ exp −c
t−τ
|x − y|2
exp −a
t
.
Combining (3.12) and (3.13), we have
(3.14)
Γa (x, t; z, τ )Γb (z, τ ; y, 0) ≤
1
ρn/2
Γc (x, t; z, τ )Γa (x, t; y, 0).
Moreover,
d(x)
min 1, √
t−τ
d(x)
1
1
1
√ ≤ √ min 1, √
√
ρ
τ
t−τ
t
and so
(3.15)
1
d(x)
d(y)
d(z)
1
√
w ≤ min 1, √
min 1, √
min 1, √
.
ρ
t−τ
t−τ
t
t
Combining (3.9), (3.14) and (3.15), we obtain
γa (x, t; z, τ )ψb (z, τ ; y, 0) ≤
1
ρn/2+1
ψc (x, t; z, τ )γa (x, t; y, 0),
which ends the proof.
Replacing γa by ψa in Lemma 3.2 and following the same manner of proof, we also obtain
Lemma 3.3. Let 0 < a < b. Then for any 0 < c < min(a, b − a), there exists a constant
C2 = C2 (a, b, c) > 0 such that for all x, y, z ∈ D, s < τ < t,
h
i
ψa (x, t; z, τ )ψb (z, τ ; y, s)
≤ C2 ψc (x, t; z, τ ) + ψc∗ (z, τ ; y, s) .
ψa (x, t; y, s)
By simple computations we also have the following inequalities.
Lemma 3.4. For 0 < a < b < c, there exists a constant C3 = C3 (a, b, c) > 0 such that, for all
x, y ∈ D and s < t,
d2 (y)
d(y)
d2 (y)
−1
C3 min 1,
Γc (x, t; y, s) ≤
γb (x, t; y, s) ≤ C3 min 1,
Γa (x, t; y, s).
t−s
d(x)
t−s
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D IRICHLET G REEN F UNCTIONS FOR PARABOLIC O PERATORS
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4. T HE C LASSES Kcloc (Ω) AND Pcloc (Ω)
In this section we introduce general classes of drift terms ν = (ν1 , . . . , νn ) and potentials
µ which guarantee the existence and uniqueness of a continuous L-Green function G for the
initial-Dirichlet problem on Ω satisfying two-sided estimates like the ones in the unperturbed
case (Theorem 2.2).
Definition 4.1 (see [37, 40]). Let B be a locally integrable Rn -valued function on Ω. We say
that B is in the parabolic Kato class if it satisfies, for some c > 0,
(
lim
r→0
Z
t
Z
sup
(x,t)∈Ω
t−r
Γc (x, t; z, τ )
√
|B(z, τ )|dzdτ
√
t−τ
D∩{|x−z|≤ r}
)
Z s+r Z
Γc (z, τ ; y, s)
√
+ sup
|B(z, τ )|dzdτ = 0.
√
τ −s
(y,s)∈Ω s
D∩{|z−y|≤ r}
Remark 4.1.
(1) Clearly, by the compactness of Ω, if B is in the parabolic Kato class then
Z tZ
sup
(x,t)∈Ω
0
D
Γc (x, t; z, τ )
√
|B(z, τ )|dzdτ
t−τ
Z
+ sup
(y,s)∈Ω
T
s
Z
D
Γc (z, τ ; y, s)
√
|B(z, τ )|dzdτ < ∞.
τ −s
(2) In the time-independent case, the parabolic Kato class is identified to the elliptic Kato
class Kn+1 (see [4], for n ≥ 3), i.e. the class of locally integrable Rn -valued functions
B = B(x) on D satisfying
Z
lim sup
ϕ(x, z)|B(z)|dz = 0,
√
r→0 x∈D
D∩{|x−z|< r}
where
(
ϕ(x, z) =
1
|x−z|n−1
if n ≥ 2
1
1 ∨ Log |x−z|
if n = 1.
Note that if B ∈ Kn+1 , then
Z
ϕ(x, z)|B(z)|dz < ∞.
sup
x∈D
D
Definition 4.2. Let c > 0 and ν = (ν1 , . . . , νn ) with νi a signed Radon measure on Ω. We say
that ν is in the class Kcloc (Ω) if it satisfies
(4.1)
Z tZ
c
M (ν) := sup
(x,t)∈Ω
ψc (x, t; z, τ )|ν|(dzdτ )
0
D
Z
T
Z
+ sup
(y,s)∈Ω
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 36, 24 pp.
s
ψc∗ (z, τ ; y, s)|ν|(dzdτ ) < ∞,
D
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8
L OTFI R IAHI
and, for any compact subset E ⊂ Ω,
(
(4.2)
lim
r→0
Z
t
Z
sup
(x,t)∈E
t−r
ψc (x, t; z, τ )|ν|(dzdτ )
√
D∩{|x−z|≤ r}
Z
s+r
+ sup
(y,s)∈E
s
)
Z
√
ψc∗ (z, τ ; y, s)|ν|(dzdτ )
= 0.
D∩{|z−y|≤ r}
Remark 4.2.
(1) From Definitions 4.1, 4.2 and Remark 4.1.1, the class Kcloc (Ω) contains the parabolic
Kato class.
(2) In the time-independent case, Kcloc (Ω) is identified to the class Kloc (D) of signed Radon
measures ν = (ν1 , . . . , νn ) on D satisfying
Z
(4.3)
sup
ψ(x, z)|ν|(dz) < ∞,
x∈D
(4.4)
D
and, for any compact subset E ⊂ D,
Z
lim sup
ψ(x, z)|ν|(dz) = 0,
√
r→0 x∈E
D∩{|x−z|< r}
where

d(z)
1

min
1,
if n ≥ 2,

|x−z| |x−z|n−1
ψ(x, z) =

 Log 1 + d(z)
if n = 1.
|x−z|
For n ≥ 3, the class Kloc (D) was recently introduced in [13] to study the existence and
uniqueness of a continuous Green function for the elliptic operator ∆ + B(x) · ∇x with
the Dirichlet boundary condition on D.
Proposition 4.3. For all α ∈]1, 2], the drift term
|Bα (z)| =
1
∈ Kloc (D) \ Kn+1 ,
d(D)
α
d(z) Log d(z)
where d(D) is the diameter of D.
Proof. Case 1: n = 1. We will prove that Bα is in the class Kloc (D). Clearly |Bα | ∈ L∞
loc (D)
and so it satisfies (4.4). We will show that Bα satisfies (4.3). We have
Z
Z
d(z)
dz
α
ψ(x, z)|Bα (z)|dz =
Log 1 +
|x − z| d(z) Log d(D)
D
D
d(z)
Z
Z
=
. . . dz +
. . . dz
D∩(|x−z|≤d(z)/2)
(4.5)
D∩(|x−z|≥d(z)/2)
:= I1 + I2 .
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D IRICHLET G REEN F UNCTIONS FOR PARABOLIC O PERATORS
9
In the case |x − z| ≤ d(z)/2, we have 23 d(x) ≤ d(z) ≤ 2d(x), and so
Z
1
3
2d(x)
I1 ≤
·
Log 1 +
dz
(Log 2)α 2d(x) |x−z|≤d(x)
|x − z|
Z
C
2d(x)
≤
Log 1 +
dr
d(x) |r|≤d(x)
|r|
Z 1
2
(4.6)
= 2C
Log 1 +
dt = C 0 .
t
0
Moreover, by using the inequality Log(1 + t) ≤ t, for all t ≥ 0, we have
Z
dz
α
I2 ≤
D |x − z| Log d(D)
|x−z|
Z d(D)
dr
α = C 0 .
(4.7)
≤C
d(D)
0
r Log r
Combining (4.5) − (4.7), we obtain that Bα satisfies (4.3).
Now we prove that Bα does not belong to the class Kn+1 . Without loss of generality, we may
assume that D =]0, 1[. We have
−α
Log 1
Z
Z 1
d(z)
1
dz
ϕ(x, z)|Bα (z)|dz = sup
Log
sup
d(z)
|x − z|
x∈[0,1] 0
x∈D D
1−α
Z 1/2 1
1
≥
Log
dz = ∞.
z
z
0
Case 2: n ≥ 2. We will prove that Bα is in the class Kloc (D). Clearly |Bα | ∈ L∞
loc (D) and so it
satisfies (4.4). We will show that Bα satisfies (4.3). We have
Z
Z
d(z)
1
dz
α
ψ(x, z)|Bα (z)|dz =
min 1,
n−1
|x − z| |x − z|
D
D
d(z) Log d(D)
d(z)
Z
Z
=
. . . dz +
. . . dz
D∩(|x−z|≤d(z)/2)
(4.8)
D∩(|x−z|≥d(z)/2)
:= J1 + J2 .
In the case |x − z| ≤ d(z)/2, we have 32 d(x) ≤ d(z) ≤ 2d(x), and so
Z
1
3
dz
J1 ≤
α
(Log 2) 2d(x) |x−z|≤d(x) |x − z|n−1
Z d(x)
C
(4.9)
≤
dr = C.
d(x) 0
Moreover,
Z
dz
J2 ≤
d(D)
α
D |x − z|n Log
|x−z|
Z d(D)
dr
α = C 0 .
≤C
(4.10)
d(D)
0
r Log r
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 36, 24 pp.
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10
L OTFI R IAHI
Combining (4.8) − (4.10), we obtain that Bα satisfies (4.3).
Now we prove that Bα does not belong to the class Kn+1 . Without loss of generality, we may
assume that 0 ∈ ∂D. D is a C 1,1 -domain and so there exists r0 > 0 such that
D ∩ B(0, r0 ) = B(0, r0 ) ∩ {x = (x0 , xn ) : x0 ∈ Rn−1 , xn > f (x0 )},
and
∂D ∩ B(0, r0 ) = B(0, r0 ) ∩ {x = (x0 , f (x0 )) : x0 ∈ Rn−1 },
where f is a C 1,1 -function. For some ρ0 > 0 small (see [30, p. 220]) the set
V0 = {z = (z 0 , zn ) : |z 0 | < ρ0 , and 0 < zn − f (z 0 ) < r0 /4}
satisfies
D ∩ B(0, ρ0 ) ⊂ V0 ⊂ D ∩ B(0, r0 /2)
and for all z ∈ V0 , d(z) ≤ zn − f (z 0 ) ≤ Cd(z) and |f (z 0 )| ≤ C 0 |z 0 |, where C and C 0 depend
only on the C 1,1 -constant. From these observations, we have
Z
sup
ϕ(x, z)|Bα (z)|dz
x∈D D
Z
≥
ϕ(0, z)|Bα (z)|dz
V0
Z
|z|1−n
=
Log
1
d(z)
−α
dz
d(z)
V0
≥
≥
=
1
C
Z
Z
1
C0
Z
1
C0
Z
|z 0 |<ρ0
(|z 0 |2 + |zn |2 )
0<zn −f (z 0 )<r0 /4
Z
|z 0 |<ρ
1−n
2
r0 /4
Z
|z 0 |<ρ0
−f (z 0 )<r
1
zn −f (z 0 )
zn −
(|z 0 |2 + |zn − f 0 (z)|2 )
0<zn
0
Log
1−n
2
0 /4
(|z 0 |2 + r2 )
−α
f (z 0 )
Log
dzn dz 0
1
zn −f (z 0 )
−α
zn − f (z 0 )
dzn dz 0
(Log( 1r ))−α
drdz 0
r
1−n
2
0
−α Z ρ0
1
tn−2
Log
n−1 dtdr
r
(t2 + r2 ) 2
0
0
−α Z ρ0 /r
Z r0 /4 1
1
1
sn−2
= 00
Log
n−1 dsdr
C 0
r
r
(s2 + 1) 2
0
1−α
Z r0 /4 1
1
1
≥ 00
Log
dr = ∞.
C 0
r
r
1
= 00
C
Z
r0 /4
1
r
Definition 4.3 (see [38, 39]). Let V be a potential in L1loc (Ω). We say that V is in the parabolic
Kato class if it satisfies, for some c > 0,
(
Z t Z
lim sup
Γc (x, t; z, τ )|V (z, τ )|dzdτ
√
r→0
(x,t)∈Ω
t−r
D∩{|x−z|< r}
Z
+ sup
(y,s)∈Ω
s
s+r
)
Z
√
D∩{|x−z|< r}
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 36, 24 pp.
Γc (z, τ ; y, s)|V (z, τ )|dzdτ
= 0.
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D IRICHLET G REEN F UNCTIONS FOR PARABOLIC O PERATORS
11
Remark 4.4.
(1) If V is in the parabolic Kato class, then, by the compactness of Ω, we have
Z tZ
sup
Γc (x, t; z, τ )|V (z, τ )|dzdτ
(x,t)∈Ω
0
D
T
Z
Z
Γc (z, τ ; y, s)|V (z, τ )|dzdτ < ∞.
+ sup
(y,s)∈Ω
s
D
(2) In the time-independent case the parabolic Kato class is identified to the elliptic Kato
class Kn , i.e. the class of functions V = V (x) ∈ L1loc (D) satisfying
Z
lim sup
Φ(x, z)|V (z)|dz = 0,
√
r→0 x∈D
D∩(|x−z|< r)
where




Φ(x, z) =



1
|x−z|n−2
if n ≥ 3;
1
1 ∨ Log |x−z|
if n = 2;
1
if n = 1.
Note that, if V ∈ Kn , then
Z
Φ(x, z)|V (z)|dz < ∞.
sup
D
x∈D
In particular Kn ⊂ L1 (D).
Definition 4.4. Let c > 0 and µ a signed Radon measure on Ω. We say that µ is in the class
Pcloc (Ω) if it satisfies
Z tZ
d(z)
c
(4.11) N (µ) := sup
γc (x, t; z, τ )|µ|(dzdτ )
(x,t)∈Ω 0
D d(x)
Z TZ
d(z)
+ sup
γc (z, τ ; y, s)|µ|(dzdτ ) < ∞,
(y,s)∈Ω s
D d(y)
and, for any compact subset E ⊂ Ω,
(
Z Z
t
(4.12)
lim
r→0
sup
(x,t)∈E
t−r
√
D∩{|x−z|≤ r}
Γc (x, t; z, τ )|µ|(dzdτ )
Z
+ sup
(y,s)∈E
s
s+r
Z
√
D∩{|z−y|≤ r}
)
Γc (z, τ ; y, s)|µ|(dzdτ )
= 0.
Remark 4.5.
(1) From Definitions 4.3, 4.4, Remark 4.4.1 and Lemma 3.4, the class Pcloc (Ω) contains the
parabolic Kato class.
(2) In the time-independent case, Pcloc (Ω) is identified to the class P loc (D) of signed Radon
measures µ on D satisfying
Z
d(z)
(4.13)
kµk := sup
g0 (x, z)|µ|(dz) < ∞,
x∈D D d(x)
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 36, 24 pp.
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12
L OTFI R IAHI
and, for any compact subset E ⊂ D,
Z
(4.14)
lim sup
g0 (x, z)|µ|(dz) = 0.
√
r→0 x∈E
D∩{|x−z|< r}
This is clear by integrating with respect to time and using Theorem 2.1. For n ≥ 3, the
class P loc (D) is introduced in [30] to study the existence and uniqueness of a continuous
Green function with the Dirichlet boundary condition for the Schrödinger equation ∆ −
µ = 0 on bounded Lipschitz domains. For n = 2, the same results hold on regular
bounded Jordan domains (see [29]).
Proposition 4.6. For α ∈ [1, 2[, the potential
Vα (z) = d(z)−α ∈ P loc (D) \ Kn .
Proof. For n ≥ 3, this is done in [30, Corollary 4.8]. We will give the proof for n ∈ {1, 2}.
Note that for α ≥ 1, Vα ∈
/ L1 (D) (see [30, Proposition 4.7]) and so Vα ∈
/ Kn . We will prove
that Vα ∈ P loc (D).
Case 1: n = 1. Vα ∈ L∞
loc (D) and so it satisfies (4.14). We show that Vα satisfies (4.13). By
Theorem 2.1, we have
Z
Z
d2−α (z)
d(z)
p
g0 (x, z)|Vα (z)|dz ≤ C
dz
d(x)d(z)
D d(x)
D |x − z| +
Z
Z
=C
. . . dz +
. . . dz
D∩(|x−z|≤d(z)/2)
D∩(|x−z|≥d(z)/2)
:= C(I1 + I2 ).
(4.15)
In the case |x − z| ≤ d(z)/2, we have 23 d(x) ≤ d(z) ≤ 2d(x), and so
Z
1−α
I1 ≤ Cd (x)
dz
|x−z|≤d(x)
≤ 2Cd
(4.16)
2−α
(D) < ∞.
Moreover,
Z
I2 ≤ C
D∩(|x−z|≥d(z)/2)
Z
|x − z|2−α
p
dz
|x − z| + d(x)d(z)
|x − z|1−α dz
≤C
D
(4.17)
≤ C 0 d2−α (D) < ∞.
Combining (4.15) – (4.17), we obtain kVα k < ∞.
Case 2: n = 2. Vα ∈ L∞
loc (D) and so it satisfies (4.14). We show that Vα satisfies (4.13). By
Theorem 2.1, we have
Z
Z 1−α
d(z)
d (z)
d(x)d(z)
g0 (x, z)|Vα (z)|dz ≤ C
Log 1 +
dz
d(x)
|x − z|2
D d(x)
D
Z
Z
=C
. . . dz +
. . . dz
D∩(|x−z|≤d(z)/2)
(4.18)
D∩(|x−z|≥d(z)/2)
:= C(J1 + J2 ).
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 36, 24 pp.
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D IRICHLET G REEN F UNCTIONS FOR PARABOLIC O PERATORS
13
Recalling that in the case |x − z| ≤ d(z)/2, we have 23 d(x) ≤ d(z) ≤ 2d(x), and using the
inequality Log(1 + t) ≤ t, for all t ≥ 0, we have
2
Z
2d(x)
−α
dz
J1 ≤ Cd (x)
Log 1 +
|x − z|
|x−z|≤d(x)
Z
dz
1−α
≤ 4Cd (x)
|x−z|≤d(x) |x − z|
= C 0 d2−α (x)
(4.19)
≤ C 0 d2−α (D) < ∞.
Moreover, by using the inequality Log(1 + t) ≤ t, for all t ≥ 0, we also have
Z
d2−α (z)
J2 ≤ C
dz
2
D∩(|x−z|≥d(z)/2) |x − z|
Z
≤C
|x − z|−α dz
D
≤ C0
Z
d(D)
0
00 2−α
=C d
(4.20)
r1−α dr
(D) < ∞.
Combining (4.18) – (4.20), we obtain kVα k < ∞.
5. T HE L-G REEN F UNCTION FOR THE I NITIAL D IRICHLET P ROBLEM
In this section we fix a positive constant c < c1 /8, where c1 is the constant in Theorem 2.2,
and we study the existence and uniqueness of a continuous L-Green function for the initialDirichlet problem on Ω when ν and µ are in the classes Kcloc (Ω) and Pcloc (Ω), respectively. A
Borel measurable function G : Ω × Ω →]0, ∞] is called an L-Green function for the initialDirichlet problem if, for all (y, s) ∈ Ω, G(·, ·; y, s) ∈ L1loc (Ω) and satisfies

LG(·, ·; y, s) = ε(y,s)



G(·, ·; y, s) = 0 on ∂D × [s, T [
(*)



limt→s+ G(x, t; y, s) = εy ,
in the distributional sense, where ε(y,s) and εy are the Dirac measures at (y, s) and y, respectively. In particular, for all f ∈ L1 (D × [s, T [) and u0 ∈ C0 (D), the initial Dirichlet problem

Lu = f on D × [s, T [



u = 0 on ∂D × [s, T [



u(x, s) = u0 (x), x ∈ D
admits a unique weak solution (see [37] – [40]) given by
Z
Z tZ
u(x, t) =
G(x, t; y, s)u0 (y)dy +
G(x, t; z, τ )f (z, τ )dzdτ.
D
s
D
We say that the Green function G is continuous if it is continuous outside the diagonal. Our first
result is the following.
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 36, 24 pp.
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14
L OTFI R IAHI
Theorem 5.1. Let ν be in the class Kcloc (Ω) with M c (ν) ≤ c0 for some suitable constant c0 .
Then, there exists a unique continuous (L0 + ν · ∇x )-Green function G for the initial-Dirichlet
problem on Ω satisfying the estimates:
C −1 γc3 (x, t; y, s) ≤ G(x, t; y, s) ≤ C γ c21 (x, t; y, s),
for all x, y ∈ D and 0 ≤ s < t ≤ T , where C, c3 are positive constants depending on n, λ, D
and T .
To prove the theorem we need the following lemma.
Lemma 5.2. Let Θ = {(x, t; y, s) ∈ Ω × Ω : t > s}, f : Θ → R continuous, satisfying
|f | ≤ Cγ c21 , for some positive constant C and ν be in the class Kcloc (Ω). Then, the function
Z tZ
p(x, t; y, s) =
f (x, t; z, τ )∇z G0 (z, τ ; y, s) · ν(dzdτ )
s
D
is continuous on Θ.
Proof of Lemma 5.2. For simplicity we use the notation X = (x, t), Y = (y, s), Z = (z, τ )
and dZ = dzdτ . By Lemma 3.2, we have, for all (X; Y ) ∈ Θ,
Z tZ
|p|(X; Y ) ≤ C
γ c21 (X; Z)ψc1 (Z; Y )|ν|(dZ)
s
D
Z tZ
≤ Cγ c21 (X; Y )
[ψc (X; Z) + ψc∗ (Z; Y )] |ν|(dZ)
s
D
≤ CM c (ν)γ c21 (X; Y ),
and so p is a real finite valued function. Let (X0 ; Y0 ) := (x0 , t0 ; y0 , s0 ) ∈ Θ be fixed and let
r0 := δ(X0 , ∂Ω) ∧ δ(Y0 , ∂Ω) ∧ δ(X0 ; Y0 ) > 0,
where
1
δ(X0 , Y0 ) = |x0 − y0 | ∨ |t0 − s0 | 2
is the parabolic distance between X0 and Y0 . Consider the compact subsets E1 = B δ X0 , r20
and E2 = B δ Y0 , r20 . Since ν ∈ Kcloc (Ω), for ε > 0, there is r ∈ 0, r20 such that
Z Z
sup
ψc (X; Z)|ν|(dZ) < ε,
X∈E1
Bδ (X,r)
and
Z Z
sup
Y ∈E2
r
ψc∗ (Z; Y )|ν|(dZ) < ε.
Bδ (Y,r)
r
For X ∈ Bδ X0 , 4 , Y ∈ Bδ Y0 , 4 , we have
Z tZ
p(X; Y ) =
f (X; Z)∇z G0 (Z; Y ).ν(dZ)
s
D
Z Z
Z Z
Z Z
=
+
+
Bδ (X0 , r2 )
Bδ (Y0 , r2 )
Bδc (X0 , r2 )∩Bδc (Y0 , r2 )
:= p1 (X; Y ) + p2 (X; Y ) + p3 (X; Y ).
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 36, 24 pp.
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D IRICHLET G REEN F UNCTIONS FOR PARABOLIC O PERATORS
15
Clearly, for Z ∈ Bδc X0 , 2r ∩ Bδc Y0 , 2r , the function (X; Y ) → f (X; Z)∇z G0 (Z; Y ) is
continuous on Bδ X0 , 4r × Bδ Y0 , 4r and satisfies
|f |(X; Z)|∇z G0 |(Z; Y ) ≤ Cγ c41 (X0 + (0, r2 /8); Z)
≤ Cd(D)ψ c41 (X0 + (0, r2 /8); Z),
for some C = C(k0 , c1 , r, Y0 ) > 0 with
Z t0 +r2 /8 Z
ψ c41 (X0 + (0, r2 /8); Z)|ν|(dZ) ≤ M c (ν) < ∞.
0
D
r
It then follows
from
the
dominated
convergence
theorem
that
p
is
continuous
on
B
X
,
×
3
δ
0
4
r
r
r
r
Bδ Y0 , 4 . Moreover, for X ∈ Bδ X0 , 4 , Z ∈ Bδ X0 , 2 and Y ∈ Bδ Y0 , 4 , we have
|f |(X; Z)|∇z G0 |(Z; Y ) ≤ Cγ c21 (X; Z),
for some C = C(k0 , c1 , r0 ) > 0. So, for all X ∈ Bδ X0 , 4r and Y ∈ Bδ Y0 , 4r ,
Z Z
|p1 |(X; Y ) ≤ C
γ c21 (X; Z)|ν|(dZ)
Bδ (X0 , r2 )
Z Z
≤ Cd(D)
Bδ (X,r)
ψ c21 (X; Z)|ν|(dZ)
≤ Cd(D)ε.
In the same way, for X ∈ Bδ (X0 , 4r ), Z ∈ Bδ (Y0 , 2r ) and Y ∈ Bδ (Y0 , 4r ), we have
|f |(X; Z)|∇z G0 |(Z; Y ) ≤ Cψc1 (Z; Y ),
for some C = C(k0 , c1 , r0 ) > 0. So, for all X ∈ Bδ (X0 , 4r ) and Y ∈ Bδ (Y0 , 4r ),
Z Z
|p2 |(X; Y ) ≤ C
ψc1 (Z; Y )|ν|(dZ)
Bδ (Y0 , r2 )
≤ C0
Z Z
ψc∗1 (Z; Y )|ν|(dZ)
Bδ (Y,r)
0
≤ C ε.
Thus p is continuous at (X0 ; Y0 ).
Proof of Theorem 5.1. For α > 0 let
Bα = {f : Θ → R, continuous : |f | ≤ C γα , for some C ∈ R}.
For f ∈ Bα we put
kf k = sup
Θ
|f |
.
γα
Clearly, (Bα , k · k) is a Banach space. Let us define the operator Λ on B c21 by
Z tZ
Λf (x, t; y, s) =
f (x, t; z, τ )∇z G0 (z, τ ; y, s) · ν(dzdτ ),
s
D
for all f ∈ B c21 . By the estimate (ii) of Theorem 2.2, Lemma 3.2 and Lemma 5.2, Λ is a bounded
linear operator from B c21 into B c21 with kΛk ≤ k0 C1 M c (ν). Assume that k0 C1 M c (ν) < 1 and
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 36, 24 pp.
http://jipam.vu.edu.au/
16
L OTFI R IAHI
define G by
G(x, t; y, s) =

P
 (I − Λ)−1 G0 (x, t; y, s) = m≥0 Λm G0 (x, t; y, s)
 G (x, t; y, s)
0
for (x, t; y, s) ∈ Θ
for (x, t), (y, s) ∈ Ω, t ≤ s.
Thus G satisfies the integral equation:
Z tZ
G(x, t; y, s) = G0 (x, t; y, s) −
G(x, t; z, τ )∇z G0 (z, τ ; y, s) · ν(dzdτ ),
s
D
for all (x, t), (y, s) ∈ Ω, and it is continuous outside the diagonal. This integral equation implies
that G is a solution of the problem (∗). Moreover by Theorem 2.2 and Lemma 3.2, we have, for
all (x, t; y, s) ∈ Θ,
X
|G(x, t; y, s) − G0 (x, t; y, s)| ≤ k0
(k0 C1 M c (ν))m γ c21 (x, t; y, s)
m≥1
k02 C1 M c (ν)
=
γ c1 (x, t; y, s).
1 − k0 C1 M c (ν) 2
(5.1)
By taking
k0 C1 M c (ν) ≤
1
2k02 ec2
+1
≤
1
2
and recalling that
k0−1 γc2 ≤ G0 ≤ k0 γc1 ,
we get from (5.1),
G(x, t; y, s) ≤ 2k0 γ c21 (x, t; y, s),
for all (x, t; y, s) ∈ Θ, and
(5.2)
d(x)
e−c2
d(y)
1
G(x, t; y, s) ≥
min 1, √
min 1, √
n ,
2k0
t−s
t − s (t − s) 2
2
for all (x, t; y, s) ∈ Θ with |x−y|
≤ 1. Using (5.2) and the reproducing property of the Green
t−s
function G (which follows from the reproducing property of G0 ) we obtain, as in [31], the
existence of constants C, c3 > 0 such that
1
G(x, t; y, s) ≥ γc3 (x, t; y, s),
C
for all (x, t; y, s) ∈ Θ.
Corollary 5.3. Let ν ∈ Kcloc (Ω) with M c (ν) ≤ c0 and G be the (L0 + ν · ∇x )-Green function
for the initial-Dirichlet problem on Ω. Then,
|∇x G|(x, t; y, s) ≤ 2k0 ψ c21 (x, t; y, s)
for all x, y ∈ D and 0 ≤ s < t ≤ T .
Proof. By using the inequality (ii) of Theorem 2.2 and Lemma 3.3, we obtain by induction,
|Λm (∇x G0 )|(x, t; y, s) ≤ k0 (k0 C1 M c (ν))m ψ c21 (x, t; y, s),
for all x, y ∈ D, 0 ≤ s < t ≤ T and mP∈ N. Assume k0 C1 M c (ν) ≤ 1/2, the derivative with
respect to x of the Green function G = m≥0 Λm G0 is given by
X
∇x G =
Λm (∇x G0 )
m≥0
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 36, 24 pp.
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D IRICHLET G REEN F UNCTIONS FOR PARABOLIC O PERATORS
17
and satisfies
|∇x G|(x, t; y, s) ≤ 2k0 ψ c21 (x, t; y, s),
for all x, y ∈ D, 0 ≤ s < t ≤ T .
Theorem 5.4. Let ν be in the class Kcloc (Ω) with M c (ν) ≤ c0 , G be the (L0 + ν.∇x )-Green
function for the initial-Dirichlet problem on Ω and µ be a nonnegative measure in the class
Pcloc (Ω). Then, there exists a unique continuous L-Green function G for the initial-Dirichlet
problem on Ω satisfying the estimates C −1 γc4 ≤ G ≤ Cγ c41 on Θ, for some positive constants
C and c4 .
To prove the theorem we need the following lemma.
Lemma 5.5. Let f : Θ → R be a continuous function satisfying |f | ≤ Cγ c41 for some positive
constant C and µ be a nonnegative measure in the class Pcloc (Ω). Then, the function
Z tZ
q(x, t; y, s) =
G(x, t; z, τ )f (z, τ ; y, s)µ(dzdτ )
s
D
is continuous on Θ.
Proof of Lemma 5.5. For simplicity we use the notation X = (x, t), Y = (y, s), Z = (z, τ )
and dZ = dzdτ . By Lemma 3.1, we have, for all (X; Y ) ∈ Θ,
Z tZ
|q|(X; Y ) ≤ C
γ c21 (X; Z)γ c41 (Z; Y )µ(dZ)
s
D
Z tZ d(z)
d(z)
≤ Cγ c41 (X; Y )
γc (X; Z) +
γc (Z; Y ) µ(dZ)
d(y)
s
D d(x)
≤ CN c (µ)γ c41 (X; Y ),
and so q is a real finite valued function. Let (X0 ; Y0 ) := (x0 , t0 ; y0 , s0 ) ∈ Θ be fixed and let
r0 := δ(X0 , ∂Ω) ∧ δ(Y0 , ∂Ω) ∧ δ(X0 , Y0 ) > 0.
r0
r0
Consider the compact
subsets
E
=
B
X
,
and
E
=
B
Y
,
. Since µ ∈ Pcloc (Ω), for
1
δ
0
2
δ
0
2
2
r ε > 0, there is r ∈ 0, 20 such that
Z Z
sup
Γc (X; Z)µ(dZ) < ε,
X∈E1
and
Bδ (X,r)
Z Z
sup
Y ∈E2
Γc (Z; Y )µ(dZ) < ε.
Bδ (Y,r)
For X ∈ Bδ X0 , 4r , we have
Z tZ
q(X; Y ) =
G(X; Z)f (Z; Y )µ(dZ)
s
D
Z Z
Z Z
Z Z
=
+
+
Bδ (X0 , r2 )
Bδ (Y0 , r2 )
Bδc (X0 , r2 )∩Bδc (Y0 , r2 )
:= q1 (X; Y ) + q2 (X; Y ) + q3 (X; Y ).
r
c
c
For Z ∈ B
X
,
∩
B
Y
,
, the function (X; Y ) → G(X; Z)f (Z; Y ) is continuous on
0
0
δ
δ
2
2
r
r
Bδ X0 , 4 × Bδ Y0 , 4 with
r
G(X; Z)|f |(Z; Y ) ≤ Cγ c41 (X0 + (0, r2 /8); Z)γ c81 (Z; Y0 − (0, r2 /8)),
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18
L OTFI R IAHI
for some C = C(k0 , c1 , r, X0 , Y0 ) > 0 and by Lemma 3.1,
2
Z
t0 + r8
2
s0 − r8
Z
D
γ c41 (X0 + (0, r2 /8); Z)γ c81 (Z; Y0 − (0, r2 /8))µ(dZ)
≤ C0 N c (µ)γ c81 (X0 + (0, r2 /8); Y0 − (0, r2 /8)) < ∞.
r
It then follows,
from
the
dominated
convergence
theorem,
that
q
is
continuous
on
B
X
,
×
3
δ
0
4
r
r
r
r
Bδ Y0 , 4 . Moreover, for Z ∈ Bδ X0 , 2 , X ∈ Bδ X0 , 4 , Y ∈ Bδ Y0 , 4 , we have
G(X; Z)|f |(Z; Y ) ≤ CΓc (X; Z),
for some C = C(k0 , c1 , r0 ) > 0 and so
Z Z
q1 (X; Y ) ≤ C
Γc (X; Z)µ(dZ) ≤ Cε.
Bδ (X,r)
In the same way,
Z Z
q2 (X; Y ) ≤ C
Γc (Z, Y )µ(dZ) ≤ Cε.
Bδ (Y,r)
Thus q is continuous at (X0 ; Y0 ).
Proof of Theorem 5.4. Let µ be a nonnegative measure in the class Pcloc (Ω) and define the operator T µ on B c41 by
Z tZ
µ
T f (x, t; y, s) =
G(x, t; z, τ )f (z, τ ; y, s)µ(dzdτ ),
s
D
for all f ∈ B c41 . By Lemma 3.1 and Lemma 5.5, T µ is a bounded linear operator from B c41 into
B c41 with
kT µ k = T µ γ c41 ≤ 2C0 k0 N c (µ).
Its spectral radius is given by
µ
µ m
rB c1 (T ) = lim k(T ) k
4
c
1
m
m→∞
1
,
2C0 k0
µ m
= inf k(T ) k
m
µ
1
m
1
µ m c m
= inf (T ) γ 41 .
m
µ
Note that if N (µ) <
then kT k < 1 and so I + T is invertible on B c41 with k(I +
µ −1
T ) k ≤ 1. Thus, for a nonnegative measure σ in the class Pcloc (Ω) with N c (σ) < 2C10 k0 , we
have
I + T µ+σ = I + T µ + T σ = (I + T µ )[I + (I + T µ )−1 T σ ]
with k(I + T µ )−1 T σ k ≤ kT σ k < 1 and so I + T µ+σ is invertible on B c41 . From this observation
we deduce that for any nonnegative measure µ in Pcloc (Ω), the operator I + T µ is invertible on
B c41 . Let us then define the function G by

 (I + T µ )−1 G(x, t; y, s) for (x, t; y, s) ∈ Θ
G(x, t; y, s) =
 G(x, t; y, s)
for (x, t), (y, s) ∈ Ω, t ≤ s.
Then G ∈ B c41 and satisfies the integral equation:
Z tZ
G(x, t; y, s) = G(x, t; y, s) −
G(x, t; z, τ )G(z, τ ; y, s)µ(dzdτ ),
s
D
for all (x, t), (y, s) ∈ Ω. In particular, G is continuous outside the diagonal, a solution of the
problem (∗) and satisfies G ≤ Cγ c41 on Θ. Moreover, by using this upper estimate, the integral
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 36, 24 pp.
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D IRICHLET G REEN F UNCTIONS FOR PARABOLIC O PERATORS
19
equation and the arguments as in the proof of Theorem 5.1, we obtain a positive constant c4 > 0
such that G ≥ C −1 γc4 on Θ.
Theorem 5.6. Let ν be in the class Kcloc (Ω) with M c (ν) ≤ c0 , G be the (L0 + ν · ∇x )-Green
function for the initial-Dirichlet problem on Ω and µ be in the class Pcloc (Ω).
+
−
Assume that rB c1 [(I + T µ )−1 T µ ] < 1, then there exists a unique continuous L-Green
4
function G for the initial-Dirichlet problem on Ω satisfying the estimates C −1 γc4 ≤ G ≤ Cγ c41
on Θ.
Conversely, assume that there exists a unique continuous L-Green function G for the initialDirichlet problem on Ω satisfying the estimates C −1 γc4 ≤ G ≤ Cγ c41 on Θ, then rBc4 [(I +
+
−
T µ )−1 T µ ] < 1.
P
+
−
Proof. For simplicity let S = (I+T µ )−1 T µ . Since rB c1 (S) < 1, for all f ∈ B c41 , m≥0 S m f ∈
4
B c41 . Let us then define G by
 P
 m≥0 S m [(I + T µ+ )−1 G](x, t; y, s) for (x, t; y, s) ∈ Θ
G(x, t; y, s) =
 G(x, t; y, s)
for (x, t), (y, s) ∈ Ω, t ≤ s.
Thus
+
G = (I + T µ )−1 G + SG
on Θ,
which yields
−
+
(I + T µ )G = G + T µ G
and so
on Θ
Z tZ
G(x, t; y, s) = G(x, t; y, s) −
G(x, t; z, τ )G(z, τ ; y, s)µ(dzdτ ),
s
D
for all (x, t), (y, s) ∈ Ω. Using this integral equation and the same arguments as in the proof of
Theorem 5.4, G is a solution of the problem (∗), continuous outside the diagonal and satisfies
the estimates C −1 γc4 ≤ G ≤ Cγ c41 on Θ.
Conversely, assume that there exists a unique continuous L-Green function G for the initialDirichlet problem on Ω satisfying the estimates C −1 γc4 ≤ G ≤ Cγ c41 on Θ, then we have
+
G = (I + T µ )−1 G + SG
on Θ,
which implies that
G=
X
+
S m [(I + T µ )−1 G] on Θ.
m≥0
µ+ −1
By recalling that (I + T ) G is the (L0 + ν · ∇x + µ+ )-Green function for the initial-Dirichlet
+
problem on Ω which satisfies the lower bound (I + T µ )−1 G ≥ C −1 γc4 on Θ, it follows that
rBc4 (S) < 1.
Corollary 5.7. Let ν and µ be in the classes Kcloc (Ω) and Pcloc (Ω), respectively, with M c (ν) ≤
c0 and N c (µ− ) ≤ c00 for some suitable constants c0 and c00 . Then, there exists a unique continuous L-Green function G for the initial-Dirichlet problem on Ω satisfying the estimates
C −1 γc4 ≤ G ≤ Cγ c41 on Θ.
Proof. It suffices to note that for c00 ≤
1
,
2k0 C0
+
−
we have kT µ k < 1 which yields
−
−
k(I + T µ )−1 T µ k ≤ kT µ k < 1,
+
−
and so rB c1 [(I + T µ )−1 T µ ] < 1.
4
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 36, 24 pp.
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20
L OTFI R IAHI
Remark 5.8.
−
(1) Note that the condition kT µ k < 1 is sufficient for the existence of the Green function
and not necessary. More precisely, we may find a negative measure µ ∈ Pcloc (Ω) with
kT −µ k as large as we wish, however its spectral radius r(T −µ ) < 1 (see [10]).
(2) As in [31], from the estimates C −1 γc4 ≤ G ≤ Cγ c41 on Θ, we may deduce two-sided
estimates for the L-Poisson kernel on Ω which imply the equivalence of the L-harmonic
measure and the surface measure on the lateral boundary ∂D×]0, T [ of Ω.
6. G LOBAL E STIMATES FOR D IRICHLET S CHRÖDINGER H EAT K ERNELS
Despite the wide study of the behavior of Schrödinger semigroups over the last three decades
(see for example [2], [5] – [7], [14] – [17], [20] – [25], [33, 34, 36, 41, 42]), global pointwise
estimates for certain Schrödinger heat kernels on bounded smooth domains remain unknown.
In this section, we are concerned ourselves with this problem and obtained global-time estimates for heat kernels of certain subcritical Schrödinger operators on bounded C 1,1 -domains.
In particular, we rectify the heat kernel estimates given by Zhang for the Dirichlet Laplacian
[42, Theorem 1.1 (b)] with an incomplete proof. We will use the notation f ∼ h to mean that
C −1 h ≤ f ≤ Ch for some positive constant C.
Let A = A(x) be a real, symmetric, uniformly elliptic matrix with λ-Lipschitz continuous
coefficients on D. Let L0 = − div(A(x)∇x ) and g0 be the Green function with the Dirichlet
boundary condition on D. By integrating the inequality in Lemma 3.1 with respect to τ and
next with respect to t and using the fact that
Z ∞
γc (x, t; y, 0)dt ∼ Ψ(x, y) ∼ g0 (x, y),
0
we obtain the following 3g0 -Theorem valid for all dimensions n ≥ 1 (see [29] for n = 2,
[9, 26, 30] and [32] for n ≥ 3).
Lemma 6.1 (3g0 -Theorem). There exists C4 = C4 (n, λ, D) > 0 such that for all x, y, z ∈ D,
g0 (x, z)g0 (z, y)
d(z)
d(z)
≤ C4
g0 (x, z) +
g0 (z, y) .
g0 (x, y)
d(x)
d(y)
Let V = V (x) be a function in the class P loc (D) defined in Remark 4.5.2 and put L = L0 +V
with the Dirichlet boundary condition on D. By Lemma 6.1 and Theorem 9.1 in [10], we know
that when kV − k ≤ 1/4C4 , the Schrödinger operator L admits a continuous Green function g
on D comparable to g0 . In particular, L is subcritical in the sense of [18, 19, 44]. Let σ0 be the
first eigenvalue of L on D which is strictly positive and G be the Dirichlet heat kernel of L on
D (the existence of G follows from Corollary 5.7 and the reproducing property). We have the
following global-time estimates on G.
Theorem 6.2. Let V be in the class P loc (D) with kV − k ≤ c00 for some suitable constant
c00 . Then the Dirichlet heat kernel G for the Schrödinger operator L = L0 + V satisfies the
following estimates: there exist constants C, c5 , c6 > 0 depending only on n, λ, D and on V
only in terms of the quantity kV k, such that for all x, y ∈ D and t > 0,
C −1 e−σ0 t ϕc6 (x, t; y, 0) ≤ G(x, t; y, 0) ≤ C e−σ0 t ϕc5 (x, t; y, 0),
where
ϕa (x, t; y, 0) = min 1,
d(x)
√
1∧ t
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 36, 24 pp.
exp −a |x−y|2
t
d(y)
√
min 1,
,
n/2
1∧t
1∧ t
a > 0.
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D IRICHLET G REEN F UNCTIONS FOR PARABOLIC O PERATORS
21
Proof. Let h0 be the first eigenfunction normalized by kh0 k2 = 1. Clearly by the comparability
g ∼ g0 and Theorem 2.1, it follows that h0 (x) ∼ d(x). From the reproducing property of G and
the estimates
C −1 γc4 (x, t; y, 0) ≤ G(x, t; y, 0) ≤ Cγ c41 (x, t; y, 0),
for x, y ∈ D, t ∈]0, 1[ (Corollary 5.7), we have
C −t d(x)d(y) ≤ G(x, t; y, 0) ≤ C t d(x)d(y),
for all t > 0 and all x, y ∈ D; and so the semigroup e−tL of L is intrinsically ultracontractive in
the sense of [2, 5, 6, 7]. Thus, for any C > 1, there exists T > 1 such that
C −1 d(x)d(y)e−σ0 t ≤ G(x, t; y, 0) ≤ Cd(x)d(y)e−σ0 t ,
for all x, y ∈ D and t ≥ T . Combining these estimates with the finite-time estimates
C −1 γc4 (x, t; y, 0) ≤ G(x, t; y, 0) ≤ Cγ c41 (x, t; y, 0),
for x, y ∈ D, t ∈]0, T [, we clearly obtain the global-time estimates stated in Theorem 6.2.
Corollary 6.3. Let λ0 be the bottom eigenvalue of L0 on D. Then, the Dirichlet heat kernel G0
of L0 on D satisfies the following estimates: there exist constants C, c5 , c6 > 0 depending only
on n, λ and D, such that for all x, y ∈ D and t > 0,
(6.1)
C −1 e−λ0 t ϕc6 (x, t; y, 0) ≤ G0 (x, t; y, 0) ≤ C e−λ0 t ϕc5 (x, t; y, 0),
and
(6.2)
where
|∇x G0 |(x, t; y, 0) ≤ C e−λ0 t Φc5 (x, t; y, 0),
exp −a |x−y|2
t
d(y)
√
Φa (x, t; y, 0) = min 1,
,
(n+1)/2
1∧t
1∧ t
a > 0.
Proof. The estimates (6.1) are given by Theorem 6.2. We will prove (6.2). From the reproducing property of G0 , the finite-time inequality (ii) in Theorem 2.2 and the inequality
G0 ≤ Ce−λ0 t ϕc5 , c5 < c1 , we have, for all t > 2,
Z
∇x G0 (x, t; y, 0) =
∇x G0 (x, t; z, t − 1)G0 (z, t − 1; y, 0)dz,
D
and so
Z
|∇x G0 |(x, t; y, 0) ≤
|∇x G0 |(x, 1; z, 0)G0 (z, t − 1; y, 0)dz
Z
2 −λ0 (t−1)
≤k e
ψc1 (x, 1; z, 0)ϕc5 (z, t − 1; y, 0)dz
D
Z
|z−y|2
2
2 −λ0 (t−1)
≤k e
min(1, d(y))
e−c1 |x−z| e−c5 t−1 dz
D
Z
2
2 |z−y|
≤ Ce−λ0 t min(1, d(y))
e−c5 (|x−z| + t−1 ) dz
D
|x − y|2
−λ0 t
≤ Ce
min(1, d(y)) exp −c5
t
D
= Ce−λ0 t Φc5 (x, t; y, 0).
This inequality combined with the finite-time inequality (ii) of Theorem 2.2 yields the estimate
(6.2).
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 36, 24 pp.
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22
L OTFI R IAHI
The following inequalities extend the ones, proved in [13] for n ≥ 3, to all dimensions n ≥ 1.
Corollary 6.4. There exists a constant C = C(n, λ, D) > 0 such that, for all x, y, z ∈ D,
(6.3)
|∇x g0 |(x, y) ≤ Cψ(x, y),
(6.4)
g0 (x, z)|∇z g0 |(z, y)
≤ C[ψ(x, z) + ψ ∗ (z, y)]
g0 (x, y)
and
(6.5)
where
|∇x g0 |(x, z)|∇z g0 |(z, y)
≤ C[ψ(x, z) + ψ ∗ (z, y)],
ψ(x, y)

d(z)

 min 1, |x−z| |x−z|1 n−1 if n ≥ 2
ψ(x, z) = ψ ∗ (z, x) =

 Log 1 + d(z)
if n = 1.
|x−z|
Proof. Inequality (6.3) holds by integrating (6.2) of Corollary 6.3 with respect to time and using
the fact that
Z ∞
Φc5 (x, t; y, 0)dt ∼ ψ(x, y).
0
Inequality (6.4) (resp. (6.5)) holds by integrating the inequality of Lemma 3.2 (resp. Lemma
3.3) with respect to τ and next with respect to t, using the facts that
Z ∞
ψc (x, t; y, 0)dt ∼ ψ(x, y)
0
and
Z
∞
γc (x, t; y, 0)dt ∼ Ψ(x, y) ∼ g0 (x, y).
0
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