DIRICHLET GREEN FUNCTIONS FOR PARABOLIC
OPERATORS WITH SINGULAR LOWER-ORDER
TERMS
Dirichlet Green Functions
for Parabolic Operators
Lotfi Riahi
LOTFI RIAHI
Department of Mathematics
National Institute of Applied Sciences and Technology,
Charguia 1, 1080, Tunis, Tunisia
EMail: Lotfi.Riahi@fst.rnu.tn
Received:
15 March, 2006
Accepted:
10 April, 2007
Communicated by:
vol. 8, iss. 2, art. 36, 2007
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S.S. Dragomir
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2000 AMS Sub. Class.:
34B27, 35K10.
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Key words:
Green function, Parabolic operator, Initial-Dirichlet problem, Boundary behavior,
Singular potential, Singular drift term, Radon measure, Schrödinger heat kernel,
Parabolic Kato class.
Abstract:
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.
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Acknowledgements:
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.
Dirichlet Green Functions
for Parabolic Operators
Lotfi Riahi
vol. 8, iss. 2, art. 36, 2007
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Contents
1
Introduction
4
2
Notations and Known Results
6
3
Basic Inequalities
8
4
The Classes Kcloc (Ω) and Pcloc (Ω)
15
5
The L-Green Function for the Initial Dirichlet Problem
27
6
Global Estimates for Dirichlet Schrödinger Heat Kernels
40
Dirichlet Green Functions
for Parabolic Operators
Lotfi Riahi
vol. 8, iss. 2, art. 36, 2007
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1.
Introduction
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 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
Dirichlet Green Functions
for Parabolic Operators
Lotfi Riahi
vol. 8, iss. 2, art. 36, 2007
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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 L-Green 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.
Dirichlet Green Functions
for Parabolic Operators
Lotfi Riahi
vol. 8, iss. 2, art. 36, 2007
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2.
Notations and Known Results
We consider the parabolic operator
L=
∂
− 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
g−∆ (x, y) =
(b − x ∨ y)(x ∧ y − a)
b−a
for n = 1,
we have the following.
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),
Dirichlet Green Functions
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vol. 8, iss. 2, art. 36, 2007
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where
Ψ(x, y) =













d(x)d(y)|x−y|2−n
d(x)d(y)+|x−y|2
Log 1 +
if n ≥ 3;
d(x)d(y)
|x−y|2
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.
Dirichlet Green Functions
for Parabolic Operators
Lotfi Riahi
For a > 0, x, y ∈ D and s < t, let
vol. 8, iss. 2, art. 36, 2007
2
1
|x − y|
exp −a
,
n/2
(t − s)
t−s
d(x)
d(y)
γa (x, t; y, s) = min 1, √
min 1, √
Γa (x, t; y, s),
t−s
t−s
Γa (x, t; y, s) =
and
d(y)
Γa (x, t; y, s)
√
.
ψa (x, t; y, s) = ψa∗ (y, t; x, s) = min 1, √
t−s
t−s
The following estimates on the L0 -Green function G0 were recently proved in
[31].
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).
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3.
Basic Inequalities
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,
d(z)
d(z)
γa (x, t; z, τ )γb (z, τ ; y, s)
≤ 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)
Dirichlet Green Functions
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Lotfi Riahi
vol. 8, iss. 2, art. 36, 2007
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γa (x, t; z, τ )γb (z, τ ; y, 0) = wΓa (x, t; z, τ )Γb (z, τ ; y, 0),
Contents
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.
.
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Case 1. τ ∈]0, ρt]. We have
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1
1
≤
.
n/2
(t − τ )
((1 − ρ)t)n/2
Combining with the inequality
|x − z|2 |z − y|2
|x − y|2
+
≥
,
t−τ
τ
t
for all τ ∈]0, t[,
Close
we obtain
(3.2)
Γa (x, t; z, τ )Γb (z, τ ; y, 0) ≤
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(z)
d(y)
|z − y|
min 1, √
≤2
min 1, √
1+ √
d(y)
t−τ
t−τ
t−τ
d(y)
|z − y|
2 d(z)
(3.3)
≤
min 1, √
1+ √
1 − ρ d(y)
τ
t
Combining (3.1) – (3.3), we obtain, for all τ ∈]0, ρt],
2
d(z)
γc (z, τ ; y, 0)γa (x, t; y, 0)
γa (x, t; z, τ )γb (z, τ ; y, 0) ≤
n+3
(1 − ρ) 2 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
(3.4)
γa (x, t; z, τ )γb (z, τ ; y, 0) ≤ C0
where C0 = C0 (a, b, c, ρ) > 0.
d(z)
γc (z, τ ; y, 0)γa (x, t; y, 0),
d(y)
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vol. 8, iss. 2, art. 36, 2007
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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 − y|2
a + c |x − z|2
≤ exp −
exp −
1−
2
t−τ
2
t−τ
b
!
a 12 2
a − c |x − y|2
a + c |x − z|2
≤ exp −
exp −
1−
.
2
t−τ
2
(1 − ρ)t
b
(a − c) 1 −
a
b
2a(1 − ρ)
1 2
(3.6)
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2
= 1,
we obtain
|x − z|2
exp −a
t−τ
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Now taking ρ so that
Dirichlet Green Functions
for Parabolic Operators
a + c |x − z|2
|x − y|2
≤ exp −
exp −a
.
2
t−τ
t
Close
From (3.5) and (3.6), we have
(3.7)
Γa (x, t; z, τ )Γb (z, τ ; y, 0) ≤
1
ρn/2
Γ a+c (x, t; z, τ )Γa (x, t; y, 0).
2
Note that (3.7) is similar to the inequality (3.2). Then by the same method used to
prove (3.4), we obtain
(3.8)
d(z)
γa (x, t; z, τ )γb (z, τ ; y, 0) ≤ C0
γc (x, t; z, τ )γa (x, t; y, 0).
d(x)
Dirichlet Green Functions
for Parabolic Operators
Lotfi Riahi
vol. 8, iss. 2, art. 36, 2007
Combining (3.4), (3.8) and using the fact that
1 2
a 2
(a − c) 1 − b
2a(1 − ρ)
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= 1,
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we get the inequality of Lemma 3.1 with C0 = C0 (a, b, c) > 0.
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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,
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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)
γa (x, t; z, τ )ψb (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(y)
min 1, √
τ
1
√ .
τ
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Let ρ ∈]0, 1[ that will be fixed later.
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
≤
Γc (z, τ ; y, 0)Γa (x, t; y, 0)
(1 − ρ)n/2
Γa (x, t; z, τ )Γb (z, τ ; y, 0) ≤
(3.10)
Moreover, by using the same inequalities as in (3.3), we obtain
4
d(x)
d(y)
(3.11)
w≤
min 1, √
min 1, √
(1 − ρ)3/2
t
t
2
1
d(z)
|z − y|
√ .
1+ √
× min 1, √
τ
τ
τ
Combining (3.9) – (3.11) and using the inequality
2
1
2
2
,
(1 + θ) exp(−αθ ) ≤ 2 1 + √
α
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for all α, θ ≥ 0, it follows that
γa (x, t; z, τ )ψb (z, τ ; y, 0) ≤ C1 ψc∗ (z, τ ; y, 0)γa (x, t; y, 0),
with
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C1 = 8 1 + √
1
b−a−c
− n+3
2
(1 − ρ)
.
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
Close
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
2
a 1/2
b
(a − c) 1 −
a(1 − ρ)
Dirichlet Green Functions
for Parabolic Operators
= 1,
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we obtain
vol. 8, iss. 2, art. 36, 2007
2
(3.13)
exp −a
|x − z|
t−τ
2
≤ exp −c
|x − z|
t−τ
2
exp −a
|x − y|
t
.
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Combining (3.12) and (3.13), we have
(3.14)
Γa (x, t; z, τ )Γb (z, τ ; y, 0) ≤
1
ρn/2
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Γc (x, t; z, τ )Γa (x, t; y, 0).
Moreover,
d(x)
min 1, √
t−τ
d(x)
1
1
√ ≤ √ min 1, √
ρ
τ
t
√
1
t−τ
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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
1
γa (x, t; z, τ )ψb (z, τ ; y, 0) ≤ n/2+1 ψc (x, t; z, τ )γa (x, t; y, 0),
ρ
which ends the proof.
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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)
Dirichlet Green Functions
for Parabolic Operators
Lotfi Riahi
By simple computations we also have the following inequalities.
vol. 8, iss. 2, art. 36, 2007
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)
−1
C3 min 1,
Γc (x, t; y, s)
t−s
d2 (y)
d(y)
γb (x, t; y, s) ≤ C3 min 1,
Γa (x, t; y, s).
≤
d(x)
t−s
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4.
The Classes 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,
(
Z t Z
Γc (x, t; z, τ )
√
lim sup
|B(z, τ )|dzdτ
r→0 (x,t)∈Ω t−r D∩{|x−z|≤√r}
t−τ
)
Z s+r Z
Γc (z, τ ; y, s)
√
|B(z, τ )|dzdτ = 0.
+ sup
√
τ −s
(y,s)∈Ω s
D∩{|z−y|≤ r}
Remark 1.
1. Clearly, by the compactness of Ω, if B is in the parabolic Kato class then
Z tZ
sup
(x,t)∈Ω
0
Γc (x, t; z, τ )
√
|B(z, τ )|dzdτ
t−τ
D
Z TZ
Γc (z, τ ; y, s)
√
+ sup
|B(z, τ )|dzdτ < ∞.
τ −s
(y,s)∈Ω s
D
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 -
Dirichlet Green Functions
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vol. 8, iss. 2, art. 36, 2007
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valued functions B = B(x) on D satisfying
Z
ϕ(x, z)|B(z)|dz = 0,
lim sup
√
r→0 x∈D
D∩{|x−z|< r}
where
1
|x−z|n−1
(
ϕ(x, z) =
if n ≥ 2
Dirichlet Green Functions
for Parabolic Operators
1
if n = 1.
1 ∨ Log |x−z|
Lotfi Riahi
Note that if B ∈ Kn+1 , then
vol. 8, iss. 2, art. 36, 2007
Z
ϕ(x, z)|B(z)|dz < ∞.
sup
D
x∈D
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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
Z tZ
c
(4.1)
M (ν) := sup
ψc (x, t; z, τ )|ν|(dzdτ )
(x,t)∈Ω
0
T
Z
+ sup
(y,s)∈Ω
s
< ∞,
sup
(x,t)∈E
t−r
I
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√
D∩{|x−z|≤ r}
Z
+ sup
(y,s)∈E
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t
lim
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ψc∗ (z, τ ; y, s)|ν|(dzdτ )
and, for any compact subset E ⊂ Ω,
(
Z Z
r→0
JJ
D
Z
(4.2)
Contents
s
s+r
ψc (x, t; z, τ )|ν|(dzdτ )
Z
√
D∩{|z−y|≤ r}
)
ψc∗ (z, τ ; y, s)|ν|(dzdτ )
= 0.
Remark 2.
1. From Definitions 4.1, 4.2 and Remark 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
Dirichlet Green Functions
for Parabolic Operators
Lotfi Riahi
D
vol. 8, iss. 2, art. 36, 2007
and, for any compact subset E ⊂ D,
Z
(4.4)
lim sup
ψ(x, z)|ν|(dz) = 0,
√
r→0 x∈E
where
D∩{|x−z|< r}

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)| =
Title Page
1
∈ Kloc (D) \ Kn+1 ,
d(D)
α
d(z) Log d(z)
where d(D) is the diameter of D.
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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
dz
d(z)
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)
Dirichlet Green Functions
for Parabolic Operators
Lotfi Riahi
D∩(|x−z|≥d(z)/2)
:= I1 + I2 .
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
vol. 8, iss. 2, art. 36, 2007
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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
sup
ϕ(x, z)|Bα (z)|dz = sup
Log
dz
|x − z|
d(z)
x∈[0,1] 0
x∈D D
1−α
Z 1/2 1
1
≥
Log
dz = ∞.
z
z
0
Dirichlet Green Functions
for Parabolic Operators
Lotfi Riahi
vol. 8, iss. 2, art. 36, 2007
loc
Case 2: n ≥ 2. We will prove that Bα is in the class K (D). Clearly |Bα | ∈
L∞
loc (D) and so it satisfies (4.4). We will show that Bα satisfies (4.3). We have
Z
Z
1
dz
d(z)
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 .
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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
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Moreover,
Z
dz
J2 ≤
d(D)
α
|x − z|n Log |x−z|
Z d(D)
dr
α = C 0 .
≤C
d(D)
0
r Log r
D
(4.10)
Dirichlet Green Functions
for Parabolic Operators
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
Lotfi Riahi
vol. 8, iss. 2, art. 36, 2007
Title Page
D ∩ B(0, r0 ) = B(0, r0 ) ∩ {x = (x0 , xn ) : x0 ∈ Rn−1 , xn > f (x0 )},
and
0
0
0
∂D ∩ B(0, r0 ) = B(0, r0 ) ∩ {x = (x , f (x )) : x ∈ R
where f is a C
1,1
n−1
},
-function. For some ρ0 > 0 small (see [30, p. 220]) the set
0
0
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0
V0 = {z = (z , zn ) : |z | < ρ0 , and 0 < zn − f (z ) < r0 /4}
satisfies
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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
Close
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
d(z)
V0
Dirichlet Green Functions
for Parabolic Operators
dz
Lotfi Riahi
≥
≥
=
=
=
≥
1
C
Z
Z
1−n
2
Log
1
zn −f (z 0 )
−α
dzn dz 0
0)
z
−
f
(z
n
0<zn
0 /4
0
−α
1
Z
Z
Log
zn −f (z 0 )
1
0 2
0
2 1−n
2
(|z
|
+
|z
−
f
(z)|
)
dzn dz 0
n
0
0
C |z0 |<ρ0 0<zn −f (z0 )<r0 /4
zn − f (z )
Z
Z r0 /4
1 −α
(Log( r ))
1
0 2
2 1−n
2
(|z
|
+
r
)
drdz 0
0
C |z0 |<ρ0 0
r
−α Z ρ0
Z r0 /4 1
1
1
tn−2
Log
n−1 dtdr
C 00 0
r
r
(t2 + r2 ) 2
0
−α Z ρ0 /r
Z r0 /4 1
1
1
sn−2
Log
n−1 dsdr
C 00 0
r
r
(s2 + 1) 2
0
1−α
Z r0 /4 1
1
1
Log
dr = ∞.
00
C 0
r
r
|z 0 |<ρ
(|z 0 |2 + |zn |2 )
−f (z 0 )<r
vol. 8, iss. 2, art. 36, 2007
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Definition 4.4 (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 Z
t
lim
r→0
sup
(x,t)∈Ω
√
D∩{|x−z|< r}
t−r
s+r
Z
)
Z
+ sup
(y,s)∈Ω
Γc (x, t; z, τ )|V (z, τ )|dzdτ
Γc (z, τ ; y, s)|V (z, τ )|dzdτ
√
s
= 0.
Dirichlet Green Functions
for Parabolic Operators
D∩{|x−z|< r}
Lotfi Riahi
Remark 3.
vol. 8, iss. 2, art. 36, 2007
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
Title Page
Contents
D
Z
T
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
Φ(x, z)|V (z)|dz = 0,
lim sup
√
r→0 x∈D
D∩(|x−z|< r)
where
Φ(x, z) =







1
|x−z|n−2
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if n ≥ 3;
1
1 ∨ Log |x−z|
if n = 2;
1
JJ
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 4.5. 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 t Z
(4.12) lim sup
Γc (x, t; z, τ )|µ|(dzdτ )
√
r→0
(x,t)∈E
t−r
+ sup
(y,s)∈E
s
s+r Z
√
D∩{|z−y|≤ r}
Lotfi Riahi
vol. 8, iss. 2, art. 36, 2007
Title Page
Contents
D∩{|x−z|≤ r}
Z
Dirichlet Green Functions
for Parabolic Operators
)
Γc (z, τ ; y, s)|µ|(dzdτ )
= 0.
Remark 4.
1. From Definitions 4.4, 4.5, Remark 3.1 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)
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and, for any compact subset E ⊂ D,
Z
g0 (x, z)|µ|(dz) = 0.
(4.14)
lim sup
√
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]).
Dirichlet Green Functions
for Parabolic Operators
Lotfi Riahi
vol. 8, iss. 2, art. 36, 2007
Proposition 4.6. For α ∈ [1, 2[, the potential
Title Page
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
loc
Vα ∈
/ Kn . We will prove that Vα ∈ P (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
d(z)
d2−α (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)
(4.15)
:= C(I1 + I2 ).
D∩(|x−z|≥d(z)/2)
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In the case |x − z| ≤ d(z)/2, we have 32 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)
vol. 8, iss. 2, art. 36, 2007
Title Page
D
≤ C 0 d2−α (D) < ∞.
Contents
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)
Lotfi Riahi
|x − z|1−α dz
≤C
(4.17)
Dirichlet Green Functions
for Parabolic Operators
D∩(|x−z|≥d(z)/2)
:= C(J1 + J2 ).
Recalling that in the case |x − z| ≤ d(z)/2, we have 23 d(x) ≤ d(z) ≤ 2d(x), and
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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)
Dirichlet Green Functions
for Parabolic Operators
≤ C 0 d2−α (D) < ∞.
Lotfi Riahi
vol. 8, iss. 2, art. 36, 2007
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
Z d(D)
≤ C0
r1−α dr
(4.20)
0
00 2−α
=C d
(D) < ∞.
Combining (4.18) – (4.20), we obtain kVα k < ∞.
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5.
The L-Green Function for the Initial Dirichlet Problem
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 initial-Dirichlet 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 initial-Dirichlet 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 ,
Dirichlet Green Functions
for Parabolic Operators
Lotfi Riahi
vol. 8, iss. 2, art. 36, 2007
Title Page
Contents
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.
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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 .
Dirichlet Green Functions
for Parabolic Operators
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,
Lotfi Riahi
vol. 8, iss. 2, art. 36, 2007
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where
1
δ(X0 , Y0 ) = |x0 − y0 | ∨ |t0 − s0 | 2
is the parabolic
distance between X0 and Y0 . Consider the compact subsetsE1 =
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
and
Bδ (X,r)
Lotfi Riahi
Z Z
ψc∗ (Z; Y
sup
Y ∈E2
)|ν|(dZ) < ε.
Bδ (X0 , r2 )
Bδ (Y0 , r2 )
Bδc (X0 , r2 )∩Bδc (Y0 , r2 )
:= p1 (X; Y ) + p2 (X; Y ) + p3 (X; Y ).
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) < ∞.
D
vol. 8, iss. 2, art. 36, 2007
Bδ (Y,r)
For X ∈ Bδ X0 , 4r , Y ∈ Bδ Y0 , 4r , we have
Z tZ
p(X; Y ) =
f (X; Z)∇z G0 (Z; Y ).ν(dZ)
Zs Z D
Z Z
Z Z
=
+
+
0
Dirichlet Green Functions
for Parabolic Operators
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It then follows
from the dominated convergence theorem that p3 is continuous
on
Bδ X0 , 4r × Bδ Y0 , 4r . Moreover, for X ∈ Bδ X0 , 4r , Z ∈ Bδ X0 , 2r and
Y ∈ Bδ Y0 , 4r , 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
γ c21 (X; Z)|ν|(dZ)
|p1 |(X; Y ) ≤ C
Bδ (X0 , r2 )
Z Z
≤ Cd(D)
ψ c21 (X; Z)|ν|(dZ)
Dirichlet Green Functions
for Parabolic Operators
Lotfi Riahi
vol. 8, iss. 2, art. 36, 2007
Bδ (X,r)
Title Page
≤ Cd(D)ε.
Contents
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 )
Z Z
0
≤C
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}.
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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 define G by

P
 (I − Λ)−1 G0 (x, t; y, s) = m≥0 Λm G0 (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.
0
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
=
1
Lotfi Riahi
vol. 8, iss. 2, art. 36, 2007
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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)
(5.1)
Dirichlet Green Functions
for Parabolic Operators
m≥1
k02 C1 M c (ν)
γ c1 (x, t; y, s).
− k0 C1 M c (ν) 2
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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),
Dirichlet Green Functions
for Parabolic Operators
G(x, t; y, s) ≤ 2k0 γ c21 (x, t; y, s),
Lotfi Riahi
for all (x, t; y, s) ∈ Θ, and
vol. 8, iss. 2, art. 36, 2007
−c2
(5.2)
G(x, t; y, s) ≥
d(x)
e
min 1, √
2k0
t−s
d(y)
min 1, √
t−s
1
n ,
(t − s) 2
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2
for all (x, t; y, s) ∈ Θ with |x−y|
≤ 1. Using (5.2) and the reproducing property
t−s
of the Green 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
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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 .
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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),
c
for all x, y ∈ D, 0 ≤ s < t ≤ T and m ∈ N. Assume
1 M (ν) ≤ 1/2, the
P k0 Cm
derivative with respect to x of the Green function G = m≥0 Λ G0 is given by
X
∇x G =
Λm (∇x G0 )
m≥0
Dirichlet Green Functions
for Parabolic Operators
Lotfi Riahi
vol. 8, iss. 2, art. 36, 2007
and satisfies
|∇x G|(x, t; y, s) ≤ 2k0 ψ c21 (x, t; y, s),
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for all x, y ∈ D, 0 ≤ s < t ≤ T .
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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
is continuous on Θ.
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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 E1 = B
X
,
and
E
=
B
Y
,
. Since µ ∈
2
δ
0
δ
0
2
2
r0
loc
Pc (Ω), for ε > 0, there is r ∈ 0, 2 such that
Z Z
sup
Γc (X; Z)µ(dZ) < ε,
X∈E1
and
Bδ (X,r)
sup
Γc (Z; Y )µ(dZ) < ε.
Bδ (Y,r)
r
For X ∈ Bδ X0 , 4 , we have
Z tZ
q(X; Y ) =
G(X; Z)f (Z; Y )µ(dZ)
Zs Z D
Z Z
Z Z
+
+
=
Bδ (X0 , r2 )
Lotfi Riahi
vol. 8, iss. 2, art. 36, 2007
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Z Z
Y ∈E2
Dirichlet Green Functions
for Parabolic Operators
Bδ (Y0 , r2 )
Bδc (X0 , r2 )∩Bδc (Y0 , r2 )
:= q1 (X; Y ) + q2 (X; Y ) + q3 (X; Y ).
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For Z ∈ Bδc X0 , 2r ∩ Bδc Y0 , 2r , the function (X; Y ) → G(X; Z)f (Z; Y ) is continuous on Bδ X0 , 4r × Bδ Y0 , 4r with
G(X; Z)|f |(Z; Y ) ≤ Cγ c41 (X0 + (0, r2 /8); Z)γ c81 (Z; Y0 − (0, r2 /8)),
for some C = C(k0 , c1 , r, X0 , Y0 ) > 0 and by Lemma 3.1,
2
Z
t0 + r8
2
s0 − r8
Z
D
Dirichlet Green Functions
for Parabolic Operators
γ c41 (X0 + (0, r2 /8); Z)γ c81 (Z; Y0 − (0, r2 /8))µ(dZ)
Lotfi Riahi
c
2
2
≤ C0 N (µ)γ (X0 + (0, r /8); Y0 − (0, r /8)) < ∞.
c1
8
It then follows,
from the
on
dominated convergence theorem,
that q3 is continuous
Bδ X0 , 4r × Bδ Y0 , 4r . Moreover, for Z ∈ Bδ X0 , 2r , X ∈ Bδ X0 , 4r , Y ∈
Bδ Y0 , 4r , 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)
vol. 8, iss. 2, art. 36, 2007
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In the same way,
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Z Z
q2 (X; Y ) ≤ C
Γc (Z, Y )µ(dZ) ≤ Cε.
Bδ (Y,r)
Thus q is continuous at (X0 ; Y0 ).
Close
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
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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
m→∞
4
c
1
,
2C0 k0
1
m
µ m
= inf k(T ) k
m
µ
1
m
1
µ m c m
= inf (T ) γ 41 .
Dirichlet Green Functions
for Parabolic Operators
m
µ
Note that if N (µ) <
then kT k < 1 and so I + T is invertible on B c41 with
µ −1
k(I + T ) k ≤ 1. Thus, for a nonnegative measure σ in the class Pcloc (Ω) with
N c (σ) < 2C10 k0 , we have
Lotfi Riahi
vol. 8, iss. 2, art. 36, 2007
Title Page
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
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estimate, the integral 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
4
L-Green 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 initial-Dirichlet problem on Ω satisfying the estimates C −1 γc4 ≤ G ≤ Cγ c41 on
+
−
Θ, then rBc4 [(I + T µ )−1 T µ ] < 1.
−
+
Proof. For simplicity let S = (I + T µ )−1 T µ . Since rB c1 (S) < 1, for all f ∈
4
P
B c41 , m≥0 S m f ∈ 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
Dirichlet Green Functions
for Parabolic Operators
Lotfi Riahi
vol. 8, iss. 2, art. 36, 2007
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+
G = (I + T µ )−1 G + SG
on Θ,
which yields
−
+
(I + T µ )G = G + T µ G
on Θ
and so
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
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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 initial-Dirichlet 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,
Dirichlet Green Functions
for Parabolic Operators
which implies that
Lotfi Riahi
G=
X
+
S m [(I + T µ )−1 G]
vol. 8, iss. 2, art. 36, 2007
on Θ.
m≥0
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+
By recalling that (I +T µ )−1 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
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Remark 5.
−
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 twosided 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.
Dirichlet Green Functions
for Parabolic Operators
Lotfi Riahi
vol. 8, iss. 2, art. 36, 2007
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6.
Global Estimates for Dirichlet Schrödinger Heat Kernels
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),
Dirichlet Green Functions
for Parabolic Operators
Lotfi Riahi
vol. 8, iss. 2, art. 36, 2007
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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 4.2 and
put L = L0 + V with the Dirichlet boundary condition on D. By Lemma 6.1 and
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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,
Dirichlet Green Functions
for Parabolic Operators
Lotfi Riahi
vol. 8, iss. 2, art. 36, 2007
Title Page
C −1 e−σ0 t ϕc6 (x, t; y, 0) ≤ G(x, t; y, 0) ≤ C e−σ0 t ϕc5 (x, t; y, 0),
Contents
where
d(x)
√
ϕa (x, t; y, 0) = min 1,
1∧ t
d(y)
√
min 1,
1∧ t
exp
2
−a |x−y|
t
1 ∧ tn/2
,
a > 0.
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),
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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)
|∇x G0 |(x, t; y, 0) ≤ C e−λ0 t Φc5 (x, t; y, 0),
Dirichlet Green Functions
for Parabolic Operators
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vol. 8, iss. 2, art. 36, 2007
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where
exp −a |x−y|2
t
d(y)
√
Φa (x, t; y, 0) = min 1,
,
(n+1)/2
1∧t
1∧ t
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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,
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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)
ψc1 (x, 1; z, 0)ϕc5 (z, t − 1; y, 0)dz
≤k e
D
Z
|z−y|2
2
2 −λ0 (t−1)
e−c1 |x−z| e−c5 t−1 dz
≤k e
min(1, d(y))
D
Z
2
2 |z−y|
−c
e 5 (|x−z| + t−1 ) dz
≤ Ce−λ0 t min(1, d(y))
D
|x − y|2
−λ0 t
≤ Ce
min(1, d(y)) exp −c5
t
−λ0 t
= Ce
Φc5 (x, t; y, 0).
D
This inequality combined with the finite-time inequality (ii) of Theorem 2.2 yields
the estimate (6.2).
The following inequalities extend the ones, proved in [13] for n ≥ 3, to all dimensions n ≥ 1.
Dirichlet Green Functions
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Lotfi Riahi
vol. 8, iss. 2, art. 36, 2007
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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)
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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).
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vol. 8, iss. 2, art. 36, 2007
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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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References
[1] D.G. ARONSON, Nonnegative solutions of linear parabolic equations, Annali
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