Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 76, pp. 1–11.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

HEAT KERNEL ESTIMATES FOR FOURTH-ORDER

NON-UNIFORMLY ELLIPTIC OPERATORS WITH

NON-STRONGLY CONVEX SYMBOLS

GERASSIMOS BARBATIS, PANAGIOTIS BRANIKAS

Abstract. We obtain heat-kernel estimates for fourth-order non-uniformly
elliptic operators in two dimensions. Contrary to existing results, the operators

considered have symbols that are not strongly convex. This entails certain

difficulties as it is known that, as opposed to the strongly convex case, there is
no absolute exponential constant. Our estimates involve sharp constants and

Finsler-type distances that are induced by the operator symbol. The main

result is based on two general hypotheses, a weighted Sobolev inequality and
an interpolation inequality, which are related to the singularity or degeneracy

of the coefficients.

1. Introduction

Let Ω be a planar domain and let

Hu = ∂2
x1

(
α(x)∂2

x1
u
)

+ 2∂2
x1x2

(
β(x)∂2

x1x2
u
)

+ ∂2
x2

(
γ(x)∂2

x2
u
)

(1.1)

be a fourth-order, self-adjoint, uniformly elliptic operator in divergence form on Ω
with measurable coefficients satisfying Dirichlet boundary conditions on ∂Ω. It has
been shown by Davies [8] that H has a continuous heat kernel G(x, x′, t) which
satisfies the Gaussian-type estimate

|G(x, x′, t)| ≤ c1t−1/2 exp
(
− c2

|x− x′|4/3

t1/3
+ c3t

)
, (1.2)

for some positive constants c1, c2, c3 and all t > 0 and x, x′ ∈ Ω. Indeed [8]
deals with the more general case of an operator of order 2m acting on a domain
in Rn, n < 2m. The study of fundamental solutions is central in the theory of
linear parabolic PDEs. For more results on heat kernel estimates for higher-order
operators we refer to [6, 7, 9, 10, 11, 13, 15, 16]. See also [14, 17] for related results
specific to fourth-order operators.

A sharp version of the Gaussian estimate (1.2) is obtained in [3] where it was
proved that

|G(x, x′, t)| ≤ cεt−1/2 exp
{
−
(3 3
√

2

16
− cθ − ε

)dM (x, x′)4/3

t1/3
+ cε,M t

}
, (1.3)

2020 Mathematics Subject Classification. 35K40, 47D06, 35K65, 35K67.
Key words and phrases. Heat kernel estimates; higher order operators;

singular-degenerate coefficients.
©2022. This work is licensed under a CC BY 4.0 license.

Submitted November 4, 2021. Published November 18, 2022.

1



2 G. BARBATIS, P. BRANIKAS EJDE-2022/76

for arbitrary ε and M positive. Here θ ≥ 0 is a constant that is related to the
regularity of the coefficients and dM (x, x′), M > 0, is a family of Finsler-type
distances on Ω which is monotone increasing and converges as M → +∞ to a
limit Finsler distance d(x, x′). The sharpness follows by comparing against the
short time asymptotics obtained in [12] for equations with constant coefficients and

which involve precisely the constant 3 3
√

2/16; we refer to [5] for a more detailed
discussion of the distance function d(x, x′).

An important assumption for both the Gaussian esimate (1.3) and for the cor-
responding asymptotic estimate of [12] is the strong convexity of the symbol

A(x, ξ) = α(x)ξ4
1 + 2β(x)ξ2

1ξ
2
2 + γ(x)ξ4

2 , x ∈ Ω , ξ ∈ R2 , (1.4)

of the operator H. The notion of strong convexity was introduced in [12] and it
applies to operators of order 2m acting on Rd which have constant coefficients. In
our context the strong convexity of the symbol (1.4) amounts to

0 ≤ β(x) ≤ 3
√
α(x)γ(x) , x ∈ Ω . (1.5)

We note that (1.5) is the assumption made for the heat kernel estimates of [3]; the
requirement for the short time asymptotics of the constant coefficient equation in
[12] is 0 < β < 3

√
αγ.

In the recent article [5] sharp Gaussian estimates where obtained for the heat
kernel of the operator (1.1) without the strong convexity assumption. Short time
asymptotics were also obtained from which follows in particular that there is no
absolute sharp exponential constant but instead the best constant depends on the
range of the function

Q(x) =
β(x)√
α(x)γ(x)

, x ∈ Ω . (1.6)

Our aim in the present article is to extend the estimates of [5] to the case where
the operator H is not uniformly elliptic and/or is not self-adjoint; in particular a
sharp exponential constant is obtained. Concerning the singularity or degeneracy,
we assume that H is locally uniformly elliptic and that there is a positive weight
function w(x) that controls in a suitable sense the behaviour of the coefficients of the
operator. Our main assumption consists of two general conditions (H1) and (H2)
on w(x), a weighted Sobolev inequality and a weighted interpolation inequality.
These conditions were introduced in [2] in order to obtain (non-sharp) Gaussian
estimates for non-uniformly elliptic self-adjoint operators. Besides conditions (H1)
and (H2) we shall assume that the symbol A(x, ξ) is close in an appropriate sense
to a certain class of a “good” symbols induced by w(x). These symbols correspond
to operators which additionally are self-adjoint and their coefficients are locally
Lipschitz, with the behavior near ∂Ω (or at infinity) being controlled by the weight
w(x). The estimates obtained herein complement analogous estimates in [4] where
non-uniformly elliptic operators with strongly convex symbol were considered. The
sharpness of the exponential constant σ∗ in our Gaussian estimate follows from the
asymptotic estimates of [5].

The proof is based on Davies’ exponential perturbation method. One has to
consider three different regimes depending on the values taken by the function
Q(x), namely 0 ≤ Q(x) ≤ 3 (the strongly convex regime), Q(x) ≤ 0 and Q(x) ≥ 0.
While the operator H may be singular or degenerate, our assumptions guarantee
that the function Q(x) is bounded away from zero and infinity, which is crucial for
the implementation of the method.



EJDE-2022/76 HEAT KERNEL ESTIMATES 3

2. Heat kernel estimates

2.1. Setting and statement of main theorem. Let Ω ⊂ R2 be open and con-
nected. We consider a differential operator H on L2(Ω) (complex-valued functions)
given formally by

Hu(x) = ∂2
x1

(
α(x)∂2

x1
u
)

+ 2∂2
x1x2

(β(x)∂2
x1x2

u) + ∂2
x2

(
γ(x)∂2

x2
u
)
, (2.1)

where α, β and γ are complex-valued, locally bounded functions on Ω. In case
Ω 6= R2 we impose Dirichlet boundary conditions on ∂Ω. The operator H is defined
by means of the quadratic form

Q(u) =

∫
Ω

{
α(x)|ux1x1

|2 + 2β(x)|ux1x2
|2 + γ(x)|ux2x2

|2
}
dx,

defined initially on C∞c (Ω). We assume that there exists a positive weight w(x)
with w±1 ∈ L∞loc(Ω) that controls the functions α(x), β(x), γ(x) in the following
sense: First, it holds

|α(x)| ≤ cw(x), |β(x)| ≤ cw(x), |γ(x)| ≤ cw(x), x ∈ Ω, (2.2)

for some c > 0 and second, the weighted G̊arding inequality

ReQ(u) ≥ c
∫

Ω

w(x)|∇2u|2 dx, u ∈ C∞c (Ω),

is valid for some c > 0 (here ∇2u denotes the vector whose components are the
second-order partial derivatives of u). This implies [1, Theorem 7.12] an analogous
inequality for the symbol A(x, ξ) of H, namely

ReA(x, ξ) ≥ cw(x)|ξ|4 , x ∈ Ω , ξ ∈ R2.

The quadratic form Q is closable and the domain of the closure is a weighted
Sobolev space which we denote by H2

w,0(Ω). We retain the same symbol, Q, for the

closure of the above form and define H the associated accretive operator on L2(Ω),
so that 〈Hf, f〉 = Q(u), f ∈ Dom(H), and Hu is given by (2.1) in the weak sense.

We make two assumptions on the weight w(x), a weighted Sobolev inequality
and a weighted interpolation inequality:

(H1) There exist s ∈ [ 1
2 , 1] and c > 0 such that

‖u‖∞ ≤ c[ReQ(u)]
s
2 ‖u‖1−s2 , u ∈ C∞c (Ω).

(H2) There exists a constant c > 0 such that∫
Ω

w
1
2 |∇u|2 dx ≤ ε

∫
Ω

w|∇2u|2 dx+ cε−1

∫
Ω

|u|2 dx,

for all 0 < ε < 1 and all u ∈ C∞c (Ω).

Both (H1) and (H2) are satisfied when H is uniformly elliptic, in which case the
best value for the exponent s is s = 1/2, showing that in the general case we cannot
expect any value that is smaller than 1/2; in particular, (H1) is valid with s = 1/2
if w(x) is bounded away from zero. We refer to [2] for a more detailed discussion
of these conditions, including examples where they are both valid.

We note that condition (H2) implies that for any k, l with 0 ≤ k, l ≤ 2, k+ l < 4,
there exists a constant c > 0 such that

(1 + λ4−k−l)

∫
Ω

w
k+l
4 |∇ku| |∇lu| dx ≤ ε ReQ(u) + cε−

k+l
4−k−l (1 + λ4)‖u‖22, (2.3)



4 G. BARBATIS, P. BRANIKAS EJDE-2022/76

for all ε ∈ (0, 1), λ > 0 and all u ∈ C∞c (Ω). Indeed, for λ = 1, (2.3) is a consequence
of (H2) and the Cauchy-Schwarz inequality; the case λ < 1 follows trivially from
the case λ = 1; finally, writing (2.3) for λ = 1 and replacing ε by ελk+l−4 we obtain
the result for λ > 1.

We define the weighted Sobolev space

W 1,∞
w (Ω) = {u ∈W 1,∞

loc (Ω) : ∃c ≥ 0 : |u(x)| ≤ cw(x), |∇u(x)| ≤ cw(x)3/4, x ∈ Ω}.

Definition 2.1. We say that the symbol A(x, ξ) lies in Gw if the functions α(x),
β(x), γ(x) are real-valued and belong in W 1,∞

w (Ω).

We think of Gw as a class of “good” symbols. By assumption (2.2) the last
condition holds true if and only if

|∇α(x)|+ |∇β(x)|+ |∇γ(x)| ≤ cw(x)3/4 , x ∈ Ω .

To state our main result we need some more definitions. We first set

Ew =
{
φ ∈ C2(Ω) ∩ L∞(Ω) : φ is real valued, there exist c > 0 such that

|∇φ| ≤ cw−1/4, |∇2φ| ≤ cw−1/2
}
.

In case where the symbol A(x, ξ) belongs in Gw (so in particular it is real-valued)
we additionally define for any M > 0 the subclass

EA,M =
{
φ ∈ Ew : A(x,∇φ(x)) ≤ 1, |∇2φ(x)| ≤M w(x)1/2, x ∈ Ω

}
;

our Gaussian estimates will be expressed in terms of the distance

dM (x, x′) = sup
{
φ(x′)− φ(x) : φ ∈ EA,M

}
for arbitrarily large (but finite) M ; we note that as M → +∞ this converges to

d(x, x′) = sup{φ(x′)− φ(x) : φ ∈ Lip(Ω) , A(y,∇φ(y)) ≤ 1 , a.e. y ∈ Ω}.
The domain Ω is essentially partitioned in three components depending on the
values of the bounded function Q(x) (cf. (1.6)). In particular, assuming always
that the symbol A(x, ξ) belongs in Gw, we define the locally Lipschitz functions

k(x) =


8 1−Q(x)

(1+Q(x))2 , if Q(x) ≤ 0,

8, if 0 ≤ Q(x) ≤ 3,

Q(x)2 − 1, if Q(x) ≥ 3,

and

σ(x) =
3

4

( 1

4k(x)

)1/3

=


3

8·41/3

(1+Q(x))2/3

(1−Q(x))1/3
, if Q(x) ≤ 0,

3
8·41/3 , if 0 ≤ Q(x) ≤ 3,

3
44/3 (Q(x)2 − 1)−1/3, if Q(x) ≥ 3.

We also set

k∗ = sup
x∈Ω

k(x) and σ∗ = inf
x∈Ω

σ(x) =
3

4
·
( 1

4k∗

)1/3

.

In the general case where the symbol does not belong in Gw we denote by θ the
following weighted distance of the symbol A(x, ξ) from Gw,

θ = inf
Ã∈Gw

sup
Ω

max
|ξ|=1

∣∣A(x, ξ)− Ã(x, ξ)
∣∣

w(x)
.

We shall think of θ as a small number.



EJDE-2022/76 HEAT KERNEL ESTIMATES 5

We now state our main result; the constants cε, cε,M may also depend on the
operator H.

Theorem 2.2. Assume that (H1) and (H2) are satisfied.

(a) Assume that the symbol A(x, ξ) belongs in Gw. Then for all ε ∈ (0, 1) and
all M large there exist cε, cε,M <∞ such that

|G(x, x′, t)| ≤ cεt−s exp
{
− (σ∗ − ε)

dM (x, x′)4/3

t1/3
+ cε,M t

}
, (2.4)

for all x, x′ ∈ Ω and t > 0.
(b) If A(x, ξ) does not belong Gw then there exists c > 0 such that for all

ε ∈ (0, 1) and all M large there exist cε, cε,M <∞ such that

|G(x, x′, t)| ≤ cεt−s exp
{
− (σ∗ − cθ − ε)

dM (x, x′)4/3

t1/3
+ cε,M t

}
,

for all x, x′ ∈ Ω and t > 0; here σ∗ and dM (x, x′) are defined as above

corresponding to a symbol Ã(x, ξ) in Gw for which |A(x, ξ) − Ã(x, ξ)| ≤
2θw(x)|ξ|4, x ∈ Ω, ξ ∈ R2.

Remark 2.3. (1) It follows from the asymptotic estimates obtained in [5] that the
constant σ∗ is the best possible.

(2) In case (b) one could define the exponential constant σ∗ and the distance

dM (x, x′) using the symbol A(x, ξ) rather than Ã(x, ξ). The resulting estimate
would be comparable to the one in the theorem; such differences are anyway ab-
sorbed in the term cθ in the exponential and we prefer to used Ã(x, ξ) for the
definition of these quantities since otherwise the proofs would be longer.

2.2. Proof of Theorem 2.2. As already mentioned, the proof makes use of Davies’
perturbative argument [8]. It follows from hypothesis (H2) that for any ψ ∈ Ew the
(multiplication) operator eψ leaves the Sobolev space H2

w,0(Ω) invariant so we may

define a sesquilinear form Qψ on H2
w,0(Ω) by Qψ(u) = Q(eψu, e−ψu); here

Q(u, v) =

∫
Ω

{
α(x)ux1x1

vx1x1
+ 2β(x)ux1x2

vx1x2
+ γ(x)ux2x2

vx2x2

}
dx

is the sesquilinear form associated to Q(·), hence

Qψ(u) =

∫
Ω

[
α(x)(eψu)x1x1(e−ψu)x1x1 + 2β(x)(eψu)x1x2(e−ψu)x1x2

+ γ(x)(eψu)x2x2
(e−ψu)x2x2

]
dx.

(2.5)

We shall need the following result, see [4, Proposition 3.2]:

Lemma 2.4. Assume that (H1) and (H2) hold. Let ψ ∈ Ew be given and let k ∈ R
be such that

ReQψ(u) ≥ −k‖u‖22
for all u ∈ C∞c (Ω). Then for any δ ∈ (0, 1) there exists a constant cδ such that

|G(x, x′, t)| ≤ cδt−s exp
{
ψ(x)− ψ(x′) + (1 + δ)kt

}
,

for all x, x′ ∈ Ω and all t > 0.



6 G. BARBATIS, P. BRANIKAS EJDE-2022/76

We now take in (2.5) ψ = λφ where λ > 0 and φ ∈ EA,M . After expanding, the
exponentials eλφ and e−λφ cancel and we obtain that Qλφ(u) is a linear combination
of terms of the form

λs
∫

Ω

bs,γ,δ(x)DγuDδu dx, (2.6)

(multi-index notation) where s+ |γ+ δ| ≤ 4 and each function bsγδ(x) is a product
of one of the functions α(x), β(x), γ(x) and first or second order derivatives of φ(x)
(see also (2.9) below). Recalling (2.2) we see that for each such term we have

|bs,γ,δ(x)| ≤ cw(x)
|γ+δ|

4 , x ∈ Ω . (2.7)

Definition 2.5. We denote by L the space of (finite) linear combinations of terms

of the form (2.6) with s+ |γ + δ| < 4 and |bs,γ,δ(x)| ≤ cw(x)
|γ+δ|

4 .

We note that if the form (2.6) belongs in L, then by (2.3) we have for any ε > 0,

|T (u)| ≤ cλs
∫

Ω

cw(x)
|γ+δ|

4 |Dγu| |Dδu|dx

≤ ε ReQ(u) + cε−
|γ+δ|

4−|γ+δ| (1 + λ
4s

4−|γ+δ| )‖u‖22
≤ εReQ(u) + cε−3(1 + λ3)‖u‖22 .

(2.8)

We now define the quadratic form

Q1,λφ(u) =

∫
Ω

{
λ4
[
α(x)φ4

x1
+ 2β(x)φ2

x1
φ2
x2

+ γ(x)φ4
x2

]
|u|2

+ λ2
{
α(x)φ2

x1
(uux1x1 + ux1x1u− 4|ux1 |2)

+ 2β(x)
[
φx1φx2(uux1x2 + ux1x2u− ux1ux2

− ux2ux1)− (φ2
x2
|ux1 |2 + φ2

x1
|ux2 |2)

]
+ γ(x)φ2

x2
(uux2x2 + ux2x2u− 4|ux2 |2)

}
+ α(x)|ux1x1 |2 + 2β(x)|ux1x2 |2 + γ(x)|ux2x2 |2

}
dx.

(2.9)

It may be seen that Q1,λφ(u) contains precisely those terms of the form (2.6) from
the expansion of Qλφ(u) for which we have s + |γ + δ| = 4. Hence, recalling also
(2.7), the difference Qλφ(·)−Q1,λφ(·) belongs in L.

We now define the polar symbol

A(x, z, z′) = α(x)z2
1z
′2
1 + 2β(x)z1z2z

′
1z
′
2 + γ(x)z2

2z
′2
2 , x ∈ Ω, z, z′ ∈ C2 .

We note that for z = z′ = ξ ∈ R2 this reduces to the symbol A(x, ξ) of H. For
x ∈ Ω and ξ, ξ′, η ∈ R2 we also set

S(x, ξ, ξ′, η) = ReA(x, ξ + iη, ξ′ + iη) + k(x)A(x, η). (2.10)

Given φ ∈ Ew and λ > 0 we define the quadratic form Sλφ on H2
w,0(Ω) by

Sλφ(u) =
1

(2π)2

∫∫∫
Ω×R2×R2

S(x, ξ, ξ′, λ∇φ)ei(ξ−ξ
′)·xû(ξ)û(ξ′) dξ dξ′ dx.



EJDE-2022/76 HEAT KERNEL ESTIMATES 7

Lemma 2.6. Assume that the symbol A(x, ξ) lies in Gw. Let φ ∈ Ew and λ > 0.
It holds that

ReQ1,λφ(u) +

∫
Ω

k(x)A(x, λ∇φ)|u|2dx = Sλφ(u),

for all u ∈ C∞c (Ω).

Proof. This follows from (2.10) by using the relation

Dαu(x) = (2π)−1

∫
R2

(iξ)αeix·ξû(ξ)dξ

for the various terms that appear in Q1,λφ; the fact that α(x), β(x) and γ(x) are
real-valued is also used here. �

We now define for each x ∈ Ω a quadratic form Γ(x, ·) in C6 by

Γ(x, p) =



(Q+ 1)|p1|2 + (Q+ 1)|p2|2 −Q|p3|2 − 2Q|p4|2 − 2Q|p5|2

−Q(3−Q)2

(1+Q)2 |p6|2,
if − 1 < Q(x) < 0,

3−Q
3 |p1|2 + 3−Q

3 |p2|2 + Q
3 |p1 + p2|2 + 4Q

3 |p3|2,
if 0 ≤ Q(x) ≤ 3,

2(Q− 3)|p1|2 + |p2|2 + 2(Q− 1)|p3|2 + 2Q−3
Q−1 (Q+ 1)(Q2 + 3)|p4|2,

if Q(x) > 3,

for each p = (p1, . . . , p6) ∈ C6. Clearly Γ(x, ·) is positive semidefinite for each
x ∈ Ω. We denote by Γ(x, ·, ·) the corresponding sesquilinear form in C6, that
is Γ(x, p, q) is given by a formula similar to the one above with each |pk|2 being

replaced by pkqk and with |p1 + p2|2 being replaced by (p1 + p2)(q1 + q2).
Next, for x ∈ Ω and ξ, η ∈ R2 we define a vector px,ξ,η ∈ R6 by

px,ξ,η =



(
α1/2[ξ2

1 −
3−Q
1+Qη

2
1 ], γ1/2[ξ2

2 −
3−Q
1+Qη

2
2 ], α1/2ξ2

1 − γ1/2ξ2
2 , α

1/2ξ1η1

+γ1/2ξ2η2, α
1/4γ1/4(ξ1η2 + ξ2η1), α1/2η2

1 − γ1/2η2
2

)
,

if − 1 < Q(x) < 0,(
α1/2[ξ2

1 − 3η2
1 ], γ1/2[ξ2

2 − 3η2
2 ], α1/4γ1/4[ξ1ξ2 − 3η1η2], 0, 0, 0

)
,

if 0 ≤ Q(x) ≤ 3,(
α1/2ξ1η1 − γ1/2ξ2η2, α

1/2(ξ2
1 −Qη2

1) + γ1/2(ξ2
2 −Qη2

2),

α1/4γ1/4[ξ1ξ2 − Q+3
Q−1η1η2], α1/4γ1/4η1η2, 0, 0

)
,

if Q(x) > 3.

A crucial property of the form Γ(x, ·) and the vectors px,ξ,η is that

S(x; ξ, ξ, η) = Γ(x, px,ξ,η, px,ξ,η), (2.11)

for all x ∈ Ω and ξ, η ∈ R2.
We finally define a quadratic form Γλφ(·) on H2

w,0(Ω) by

Γλφ(u) =
1

(2π)2

∫∫∫
Ω×R2×R2

Γ(x, px,ξ,λ∇φ, px,ξ′,λ∇φ)ei(ξ−ξ
′)·xû(ξ)û(ξ′) dξ dξ′ dx.



8 G. BARBATIS, P. BRANIKAS EJDE-2022/76

Lemma 2.7. Assume that the symbol A(x, ξ) lies in Gw. Then the difference
Sλφ(·)− Γλφ(·) belongs to L.

Proof. We consider the difference

S(x, ξ, ξ′, η)− Γ(x, px,ξ,η, px,ξ′,η),

of the two symbols and we group together terms that have the property that if we
set ξ′ = ξ then they are similar as monomials of the variables ξ and η. Due to
(2.11) one can use integration by parts to conclude that the total contribution of
each such group belongs to L. We shall illustrate this for two particular groups,
the one consisting of terms which for ξ = ξ′ involve the monomial ξ2

1η
2
1 and those

which for ξ = ξ′ involve ξ2
1η

2
2 . For the sake of brevity we shall consider directly the

sum of the terms of both groups.
The terms of these two groups from S(x, ξ, ξ′, η) add up to

−α(x)η2
1(ξ2

1 + ξ′21 + 4ξ1ξ
′
1)− 2β(x)η2

2ξ1ξ
′
1.

The corresponding terms in Γ(x, px,ξ,η, px,ξ′,η) are

α(x)η2
1

[
(Q(x)− 3)(ξ2

1 + ξ′21 )− 2Q(x)ξ1ξ
′
1

]
− 2β(x)η2

2ξ1ξ
′
1,

if Q(x) ≤ 0,

−3α(x)η2
1(ξ2

1 + ξ′21 )− β(x)η2
2(ξ2

1 + ξ′21 ),

if 0 ≤ Q(x) ≤ 3,

α(x)η2
1

[
−Q(x)(ξ2

1 + ξ′21 ) + 2(Q(x)− 3)ξ1ξ
′
1

]
− β(x)η2

2(ξ2
1 + ξ′21 ),

if Q(x) ≥ 3.

Hence the difference of these terms in S(x, ξ, ξ′, η)− Γ(x, px,ξ,η, px,ξ′,η) is
α(x)η2

1

[
2−Q(x)

]
(ξ1 − ξ′1)2, if Q(x) ≤ 0,[

2α(x)η2
1 + β(x)η2

2

]
(ξ1 − ξ′1)2, if 0 ≤ Q(x) ≤ 3,[

α(x)(Q(x)− 1)η2
1 + β(x)η2

2

]
(ξ1 − ξ′1)2, if Q(x) ≥ 3.

This can also be written as
[
α(x)η2

1R(x) + η2
2P (x)

]
(ξ1 − ξ′1)2 where

R(x) =


2−Q(x), if Q(x) ≤ 0,

2, if 0 ≤ Q(x) ≤ 3,

Q(x)− 1, if Q(x) ≥ 3,

and P (x) =

{
0, if β(x) ≤ 0,

β(x), if β(x) ≥ 0.

Inserting this in the triple integral and recalling that η = λ∇φ we obtain that
the contribution of the above terms in the difference Sλφ(u)− Γλφ(u) is

(2π)−2

∫∫∫
Ω×R2×R2

[
α(x)R(x)φ2

x1
+ P (x)φ2

x2

]
× (ξ1 − ξ′1)2λ2ei(ξ−ξ

′)·xû(ξ)û(ξ′) dξ dξ′ dx

= λ2

∫
Ω

[
α(x)R(x)φ2

x1
+ P (x)φ2

x2

]
(−ux1x1u− uux1x1 − 2|ux1 |2)dx

= −λ2

∫
Ω

[
α(x)R(x)φ2

x1
+ P (x)φ2

x2

]
(ux1

u+ uux1
)x1
dx

= λ2

∫
Ω

[
α(x)R(x)φ2

x1
+ P (x)φ2

x2

]
x1

(ux1u+ uux1)dx,



EJDE-2022/76 HEAT KERNEL ESTIMATES 9

where we have used that the function α(x)R(x)φ2
x1

+ P (x)φ2
x2

is locally Lipschitz.
To conclude that the last expression belongs in L we must prove that (2.7) is valid,
that is

∣∣[α(x)R(x)φ2
x1

+P (x)φ2
x2

]x1

∣∣ ≤ cw(x)1/4. We shall only consider the first of
the two terms, the proof being similar for the second. Using the relations |Q(x)| ≤ c,
|∇Q(x)| ≤ cw(x)−1/4 we obtain∣∣(α(x)R(x)φ2

x1
)x1

∣∣ ≤ |αx1
R|φ2

x1
+ |αRx1

|φ2
x1

+ 2|αRφx1
φx1x1

|

≤ cw3/4w1/2 + cww−1/4w1/2 + cww−1/4Mw1/2 = cMw
1/4,

as required. �

Lemma 2.8. Assume that the symbol A(x, ξ) lies in Gw and let M > 0 be given.
Then for all φ ∈ EA,M and λ > 0 we have

ReQλφ(u) ≥ −k∗λ4 ‖u‖22 + T (u),

for some quadratic form T ∈ L and all u ∈ C∞c (Ω).

Proof. The assumption φ ∈ EA,M implies that A(x,∇φ(x)) ≤ 1, x ∈ Ω. Recalling
that the difference Qλφ(·)−Q1,λφ(·) belongs in L and using Lemmas 2.6, 2.7, and
2.8, we obtain

ReQλφ(u) = −
∫

Ω

k(x)A(x, λ∇φ) |u|2 dx+ Γλφ(u) + T (u)

≥ −k∗λ4

∫
Ω

|u|2 dx+ Γλφ(u) + T (u),

for some form T ∈ L and all u ∈ C∞c (Ω). Moreover

Γλφ(u) =
1

(2π)2

∫∫∫
Ω×R2×R2

Γ(x, px,ξ,λ∇φ, px,ξ′,λ∇φ)ei(ξ−ξ
′)·xû(ξ)û(ξ′) dξ dξ′ dx

=
1

(2π)2

∫
Ω

Γ
(
x,

∫
R2

eiξ·xû(ξ)px,ξ,λ∇φdξ,

∫
R2

eiξ
′·xû(ξ′)px,ξ′,λ∇φdξ

′
)
dx ≥ 0,

by the positive semi-definiteness of Γ; the result follows. �

Proof of Theorem 2.2. (a) We claim that for any ε and M positive there exists cε,M
(which may also depend on the operator H) such that

ReQλφ(u) ≥ −
{

(k∗ + ε)λ4 + cε,M (1 + λ3)
}
‖u‖22. (2.12)

for all λ > 0 and φ ∈ EA,M . To prove this we first recall (cf. (2.8)) that any form
T ∈ L satisfies

|T (u)| ≤ εQ(u) + cε,M (1 + λ3) ‖u‖22,
for all ε ∈ (0, 1), λ > 0 and u ∈ C∞c (Ω). Hence, since Q(u) is real, Lemma 2.8
implies

ReQλφ(u) ≥ −
{
k∗λ4 + cε,M (1 + λ3)

}
‖u‖22 − εQ(u). (2.13)

Now, considering the expansion of Qλφ already discussed and recalling (2.3) we
infer that there exists a constant cM such that for any φ ∈ EA,M and λ > 0 it holds∣∣Q(u)−Qλφ(u)

∣∣ ≤ 1

2
Q(u) + cM (λ+ λ4)‖u‖22 . (2.14)

Furthermore, we note that the dependence on M in this estimate comes from those
terms in the expansion of Qλφ that contain at least one second-order derivative of



10 G. BARBATIS, P. BRANIKAS EJDE-2022/76

φ. Since the coefficient of λ4 in the expansion only involves first derivatives of φ,
(2.14) can be improved to∣∣Q(u)−Qλφ(u)

∣∣ ≤ 1

2
Q(u) +

{
cM (λ+ λ3) + cλ4

}
‖u‖22,

which in turn implies

Q(u) ≤ 2 ReQλφ(u) +
{
cM (λ+ λ3) + cλ4

}
‖u‖22. (2.15)

Let u ∈ C∞c (Ω) be given. If ReQλφ(u) ≥ 0 then (2.12) is obviously true. If not we
then have from (2.13) and (2.15)

ReQλφ(u) ≥ −
{
k∗λ4 + cε,M (1 + λ3)

}
‖u‖22 − 2ε ReQλφ(u)

− ε
{
cM (λ+ λ3) + cλ4

}
‖u‖22

≥ −
{

(k∗ + cε)λ4 + cε,M (1 + λ3) + ε
{
cM (λ+ λ3) + cλ4

}}
‖u‖22,

and (2.12) again follows; hence the claim has been proved.
We complete the standard argument; Lemma 2.4 and (2.12) imply

|G(x, x′, t)| < cεt
−s exp

{
λ
(
φ(x)− φ(x′)

)
+ (1 + ε)

{
(k∗ + ε)λ4 + cε,M (1 + λ3)

}
t
}
,

for all ε ∈ (0, 1). Optimizing over φ ∈ EA,M yields

|G(x, x′, t)| < cεt
−s exp

{
− λdM (x, x′) + (1 + ε)

{
(k∗ + ε)λ4 + cε,M (1 + λ3)

}
t
}
.

Finally choosing λ = [dM (x, x′)/(4k∗t)]1/3 we have

−λdM (x, x′) + k∗λ4t = −σ∗
dM (x, x′)4/3

t1/3
,

and (2.4) follows.

(b) There exists a symbol Ã(x, ξ) in Gw such that

max
{
|α(x)− α̃(x)| , |β(x)− β̃(x)| , |γ(x)− γ̃(x)|

}
≤ 2θ w(x) , x ∈ Ω.

Given φ ∈ EÃ,M and λ > 0 it follows from the proof of part (a) that

Re Q̃λφ(u) ≥ −
{
k∗λ4 + cε,M (1 + λ3)

}
‖u‖22 − εReQ(u), (2.16)

for all u ∈ C∞c (Ω). Moreover it is easily seen that∣∣Qλφ(u)− Q̃λφ(u)
∣∣ ≤ cθ{ReQ(u) + λ4‖u‖22

}
. (2.17)

The argument used for (2.15) also applies to H and we thus obtain

ReQ(u) ≤ 2 ReQλφ(u) +
{
cM (λ+ λ3) + cλ4

}
‖u‖22. (2.18)

Combining (2.16), (2.17) and (2.18) we conclude that

ReQλφ(u) ≥ −
{

(k∗ + cθ + ε)λ4 + cε,M (1 + λ3)
}
‖u‖22, u ∈ C∞c (Ω),

and the argument is completed as in part (a); we omit further details. �



EJDE-2022/76 HEAT KERNEL ESTIMATES 11

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Gerassimos Barbatis

Department of Mathematics, National and Kapodistrian University of Athens, Panepis-
timioupolis, 15784 Athens, Greece

Email address: gbarbatis@math.uoa.gr

Panagiotis Branikas

Department of Mathematics, National and Kapodistrian University of Athens, Panepis-
timioupolis, 15784 Athens, Greece

Email address: pbranikas@math.uoa.gr


	1. Introduction
	2. Heat kernel estimates
	2.1. Setting and statement of main theorem
	2.2. Proof of Theorem ??

	References