Sharp heat kernel bounds for a class of parabolic
operators with singular coefficients
G. Metafune
Department of Mathematics and Physics “Ennio De Giorgi”
Università del Salento
Lecce, Italy.
Introduction
The heat equation ∂t u − ∆u = 0, u(0) = f has the explicit solution
Z
e t∆ f (x) =
p∆ (t, x, y )f (y ) dy .
RN
where the heat kernel is given by
N
p∆ (t, x, y ) = (4πt)− 2 e −
|x−y |2
4t
If the Laplacian is replaced by A = div (a(x) · ∇), λI ≤ a(x) ≤ ΛI , Aronson
proved that the heat kernel pA satisfies
N
C1 t − 2 e −c1
|x−y |2
t
N
≤ pA (t, x, y ) ≤ C2 t − 2 e −c2
|x−y |2
t
.
Two sided gaussian bounds have been proved, then, by several authors,
extending Aronson’s results to second order operators with lower order
terms. For example they are known when
A = div (a(x) · ∇) + b · ∇ − V
and V belongs to the Kato class KN , b to KN+1 (Simon, Lischevith,
Semenov, Voigt, etc..).
Roughly speaking a singularity of V , say at 0 can be at most like |x|−2+ε
and that of b like |x|−1+ε .
Non-autonomous cases are also treated and V , b are assumed to belong to
suitable non-autonomous Kato classes.
The operator
For a > 0, b, c ∈ R, we consider the elliptic operator
L = ∆ + (a − 1)
N
X
xi xj
x
b
Dij + c 2 · ∇ − 2 .
2
|x|
|x|
|x|
i,j=1
The diffusion matrix a(x) := I + (a − 1) x⊗x
has eigenvalues a with
|x|2
eigenvector x and 1 with eigenspace x ⊥ .
L invariant by rescaling, that is if Mλ u (x) := u(λx) then
L(Mλ u)(x) = λ2 Lu (λx)
or Mλ−1 ◦ L ◦ Mλ = λ2 L.
Note that L becomes
A Schrödinger Operator: a = 1, c = 0
L=∆−
b
.
|x|2
An operator with discontinuous coefficients: b = c = 0
L = ∆ + (a − 1)
N
X
xi xj
Dij
|x|2
i,j=1
well-known in elliptic regularity.
An operator with unbounded coefficients (after a change of
variable)
Jp ◦ L ◦ Jp−1 = |x|α ∆ + c̃|x|α−1
x
· ∇ − b̃|x|α−2 .
|x|
In spherical coordinates x = r ω,
∆ = Drr +
N−1
r Dr
+
∆0
,
r2
∆0 is the Laplace-Beltrami operator on S N−1 .
N
X
xi xj
x
b
L = ∆ + (a − 1)
Dij + c 2 · ∇ − 2
2
|x|
|x|
|x|
i,j=1
= aDrr +
b − ∆0
N −1+c
Dr −
.
r
r2
Lu = 0 has radial solutions
|x|−s1 ,
|x|−s2
where s1 , s2 are the roots of the indicial equation
f (s) = −as 2 + (N − 1 + c − a)s + b = 0
s1/2
N −1+c −a √
:=
∓ D,
2a
b
D := +
a
N −1+c −a
2a
2
.
Theorem (Metafune, Spina, Sobajima)
2
(i) If D = ba + N−1+c−a
< 0, then there exists f ≥ 0 such that
2a
u − Lu = f does not have any positive distributional solutions.
2
(ii) If D = ba + N−1+c−a
≥ 0 there exists a realization of L, between
2a
the minimal and the maximal operator which generates a semigroup
in Lp (RN ) iff Np ∈ (s1 , s2 + 2).
The description of the domain in Lp (RN ) depends on weighted Rellich and
Calderón-Zygmund inequalities.
Note that if a=1 and c=0
L := ∆ −
b
|x|2
Schrödinger operator
D = b + (N − 2)2 /4
and [(i)] is a result due to Baras-Goldstein who actually prove more.
The operator in L2 (RN , |x|γ dx)
We define L through a symmetric form in a weighted space. Writing
t
a(x) := I + (a − 1) x|x|·x2 ,
γ=
N−1+c
a
− N + 1, we have
L = |x|−γ div(|x|γ a ∇) −
b
.
|x|2
Let L2µ = L2 (RN , dµ) with dµ = |x|γ dx. We introduce the symmetric form
Z
a(u, v ) :=
RN
b
ha∇u, ∇v i + 2 uv dµ,
|x|
D(a) := Cc∞ (RN \ {0})
which satisfies (−Lu, v )L2µ = a(u, v ) for all u ∈ Cc∞ (RN \ {0}).
The positivity of the form is equivalent to a weighted Hardy inequality,
which is ensured (actually it is equivalent to) by the positivity of D.
If D ≥ 0, a is nonnegative, symmetric and closable in L2µ = L2 (RN , dµ).
Denoting by ã the closure of a, we can define −L as the operator
associated to ã given by
Z
2
D(L) := u ∈ D(ã) ; ∃v ∈ Lµ s.t. ã(u, w ) =
v w dµ ∀w ∈ D(a) ,
RN
−Lu := v .
The following proposition follows from the theory of symmetric forms.
Proposition If D ≥ 0, the operator −L is nonnegative, selfadjoint and
generates a positive semigroup e tL in L2µ . Moreover,
(i) Cc∞ (RN \ {0}) ,→ D(L) and for every u ∈ Cc∞ (RN \ {0})
Lu =
N
X
aij Dij u + c
i,j=1
x
b
· ∇u − 2 u
2
|x|
|x|
(ii) Defining, for s > 0, Ms f (x) := f (sx) we obtain
s 2 L = Ms−1 LMs ,
es
2 tL
= Ms−1 e tL Ms .
The following theorem summarizes the main result of this talk.
Theorem
If D ≥ 0 e t L is a positive semigroup of integral operators.
The heat kernel p with respect to the measure dµ = |y |γ dy , satisfies
p(t, x, y ) ' t
− N2
− γ2
|x|
|y |
− γ2
|x|
1
t2
− N +1+√D
2
c|x−y |2
|y |
∧1
e− t .
1 ∧ 1
t2
The constant c > 0 may differ in the upper and lower bounds.
The symbol f ' g means that there are two positive constants C1 , C2 such
that C1 f (x) ≤ g (x) ≤ C2 f (x) for every x where f , g are defined.
We note some special cases of the above bounds
√
|x|
|y |
(i) If √ ≤ 1, √ ≤ 1,
t
t
then p(t, x, y ) ' t −1−
|x|
|y |
(ii) if √ ≥ 1, √ ≥ 1,
t
t
then p(t, x, y ) ' t − 2 |y |−γ e −
N
D
|x|−s1 |y |−s1
m|x−y |2
t
.
√
√
The mixed cases when |x|/ t is small and |y |/ t large, and conversely,
must however be also considered.
An equivalent way to express our bounds is the following, where the kernel
is written with respect to the Lebesgue measure, and denoted by pL
pL (t, x, y ) ' t
− N2
|x|
1
−s1 ∧1
t2
|y |
1
−s1∗
∧1
e−
c|x−y |2
t
.
t2
where −s1 , −s1∗ are the greatest roots of the indicial equation relative to
L, L∗ . Gradient estimates are obtained from those for Dt p, Lp via
interpolation for |x| large and by the series representaion of the kernel (see
later), for small |x|.
Proposition
|∇x pL (t, x, y )| ≤ Ct
for suitable c, C > 0.
− N+1
2
|x|
1
t2
−s1 −1 ∧1
|y |
1
t2
−s1∗
c|x−y |2
∧1
e− t .
The upper estimate
Theorem (Metafune, Spina, Sobajima)
If D ≥ 0, e t L is a positive semigroup of integral operators.
The heat kernel p (with respect to the measure dµ) satisfies
p(t, x, y ) ≤ Ct
− N2
|x|
− γ2
− γ2
|y |
|x|
1
t2
− N +1+√D
2
|x−y |2
|y |
∧1
e − ct .
1 ∧ 1
t2
Remark
2 tL
= Ms−1 e tL Ms , where Ms f (x) := f (sx)
x y
− N2 − γ2
p(t, x, y ) = t
p 1, √ , √ .
t
t
By the scaling property e s
Therefore it is sufficient to prove the estimate for t = 1.
We use ”Nash” inequality to prove ultracontractivity and then Davies
method to add the Gaussian factor. First we change the undelying measure
to get rid of the potential which is unbounded from below. This is done
using a modification of the radial ”eigenfunction” of L, namely |x|−s1 .
(
γ
if |x| ≥ 1;
|x|− 2 ,
L(φ)
which satisfies
φ(x) :=
φ ≤ c0
−s
1
|x| ,
if |x| ≤ 1,
The map u ∈ L2 (RN , dµ) 7→ φ−1 u ∈ L2 (RN , φ2 dµ) yields the operator
identity
L(u) = φ L̃(φ−1 u).
−L̃ is associated to the symmetric form
b(u1 , v1 ) : = a(φu1 , φv1 )
Z L(φ)
u1 v1 φ2 dµ.
=
ha∇u1 , ∇v1 i −
φ
RN
The potential term of −L̃ is bounded from below, hence
Z
Z
0
2 2
b (u1 , u1 ) := b(u1 , u1 ) + co
|u1 | φ dµ ≥
(ha∇u1 , ∇u1 i) φ2 dµ.
RN
RN
L̃ + c0 generates e t(L̃+c0 ) = e tc0 e t L̃ in L2 (RN , φ2 dµ). We show that
e t(L̃+c0 ) is L∞ -contractive, using the classical Beurling-Deny conditions
and that e t(L̃+c0 ) is ultracontractive:
ke t(L̃+c0 ) kL(L1 ,L∞ ) ≤ ct −α ,
where α = −N/2 or α = −(N + γ + 2s1 )/2 according to D ≤ (N − 2)2 /4
or D > (N − 2)2 /4.
The main tools are Gagliardo-Nirenberg type inequalities (which we
formulate when D ≤ (N − 2)2 /4): ∃q ∈ (2, ∞], N q−2
2q < 1 s.t.
kukq ≤ cb0 (u, u)
N q−2
4q
1−N q−2
2q
kuk2
for all u ∈ D(b0 ).
By the Dunford-Pettis Theorem e t L̃ admits a kernel pL̃ (t, x, y ) with
respect to φ(y )2 dµ s.t.
Z
t L̃
(i) e f (x) =
pL̃ (t, x, y )f (y ) φ(y )2 dµ.
X
N
(ii) pL̃ (t, x, y ) ≤ ct − 2 e −c0 t .
Next we use ”Davies’s method” to add a Gaussian factor. Let
e −αΨ L̃e αΨ ,
TαΨ f = e −αΨ e t L̃ (e αΨ f )
α ∈ R, Ψ ∈ F ⊆ C ∞ (RN \ {0}) (adapted to the the degeneracy).
We prove as above that
N
pαΨ (t, x, y ) ≤ ct − 2 e −c(1+α
and using
pαΨ = e α(Ψ(y )−Ψ(x)) pL̃
we arrive at
2 )t
N
pL̃ (t, x, y ) ≤ ct − 2 e −c(1+α
2 )t
e α(Ψ(x)−Ψ(y )) .
Optimizing over α ∈ R and ψ ∈ F
N
|pL̃ (t, x, y )| ≤ ct − 2 e −c1 t e −C
|x−y |2
t
.
Coming back to L we obain from the identity
pL (t, x, y ) = φ(x)φ(y ) pL̃ (t, x, y )
and the defintion of φ, that
− γ2
pL (1, x, y ) ≤ C |x|
|y |
− γ2
h
i− N +1+√D
2
2
(1 ∧ |x|)(1 ∧ |y |)
e −c|x−y | .
The case D > (N − 2)2 /4 is similar, using another Gagliardo-Nirenberg
type inequality.
Lower estimate-1d
If N = 1, then γ = ca , D =
b
a
+
c−1 2
2a
≥ 0 and
c
b
Lu = aurr + ur − 2 u.
r
r
c
L is defined through the symmetric form in L2 (0, ∞), r a dr
Z
b(u, v ) :=
0
∞
uv
aur vr + b 2
r
c
r a dr .
c
The heat kernel (with respect to the measure s a ds ) is given by
p(t, r , s) =
rs r 2 +s 2
c−a
1
(rs)− 2a I√D
e − 4at
2at
2at
Here
Iν (r ) =
∞
r ν X
2
m=0
r 2m
1
,
m! Γ(ν + 1 + m) 2
Kν (r ) =
π I−ν (r ) − Iν (r )
2
sin πν
are the modified Bessel functions and consitute a basis of solutions of the
homogenuous equation
2
vr
ν
vrr +
−
+ 1 v = 0.
r
r2
c−a
b
c
Note that if u − (aurr + ur − 2 u) = 0 and u(r ) := r − 2a v ( √r a ), then
r
r
D
vr
vrr +
−
+ 1 v = 0.
r
r2
From the the asymptotic behavior of the modified Bessel function
 1
r − 2 e r , r → ∞
√
r
D+ 12 e
√
√
I D (r ) ≈
'
(1
∧
r
)
√
r D
r
, r →0
we deduce that
√
c
1
rs D+ 21 − |r −s|2
− 2a
p(t, r , s) ' √ (rs)
1∧
e 4t .
t
t
From 1d to Nd
∆0 is the Laplace-Beltrami on L2 (S N−1 ),
∆ = Drr +
N−1
r Dr
+
∆0
.
r2
If p(x) = r n P n (ω) is an homogeneous harmonic polynomial of degree n
then P n is a spherical harmonic of order n and
∆0 P n = −(n2 + n(N − 2))P n := −λn P n .
We set
(i) Hn = {spherical harmonics of degree n}.
(ii) {Pin , i = 1, ..., an } orthonormal basis of Hn .
Then
L2 (S N−1 ) =
∞
M
n=0
Hn
The projection of f onto Hn is given by
fn (ω) =
an Z
X
S N−1
i=1
(n)
where Zω (η) =
Pan
f (η)Pin (η) dη
n
n
i=1 Pi (ω)Pi (η)
Pin (ω)
Z
=
S N−1
f (η)Z(n)
ω (η), dη.
is the zonal harmonic of degree n.
We also recall that
sup |Zω(n) (η)| = Zω(n) (ω) =
η∈S N−1
as n → ∞.
an
|S N−1 |
≈
nN−2
,
|S N−1 |
Next we consider L2µ = L2 (RN , |x|γ dx) , γ =
the subspaces
L2n = L2 ((0, ∞), r
N−1+c
a
N−1+c
a
− N + 1 and define
dr ) ⊗ Hn
Since
Z
RN
X
2
f (r )Pin (ω) |x|γ dx =
0
i
it follows that L2µ =
Z
L∞
2
n=0 Ln ,
u=
Z
P∞
Qn (u)(r , η) :=
S N−1
∞X
|f (r )|2 r γ+N−1 dr
i
n=0 Qn (u)
where
u(r , ω)Zω(n) (η)dω.
Decomposition of the operator
We write L = aDrr +
N−1+c
Dr
r
−
b−∆0
r2
in spherical coordinates.
If uP, vP ∈ Cc∞ (RN \ {0}) ∩ L2n then
N −1+c
b + λn
L(uP) = aurr +
ur −
u P(ω) := (Ln u)(r )P(ω).
r
r2
Ln is the Bessel operator on L2 ((0, ∞), r
Z
an (u, v ) =
0
∞
N−1+c
a
b + λn
uv
aur v r +
r2
a|
L2
n
= an
=⇒
ã|
dr ) associated to the form
r
L2
n
N−1+c
a
= ãn .
dr = a(uP, vP)
Moreover, f = u(r )P(ω) ∈ D(L) if and only if u ∈ D(Ln ) and in such a
case
L u(r )P(ω) = Ln u(r ) P(ω).
It follows that e tL u(r )P(ω) = e tLn u(r ) P(ω) = e tLn f .
If f ∈ L2µ , f =
P∞
n=0 Qn (f )
then
e tL f =
∞
X
e tLn Qn f .
n=0
Writing this equality in terms of the kernels we obtain (still formally)
X
p(t, x, y ) =
pn (t, r , ρ)Z(n)
x = r ω, y = ρη
ω (η),
n≥0
.
Here
2
rρ 1
r + ρ2
− N−1+c−a
√
2a
I Dn
(r ρ)
pn (t, r , ρ) =
exp −
2at
2at
4at
is the kernel, with respect to ρ
N−1+c
a
Ln = aDrr +
dρ, of
N −1+c
b + λn
Dr −
r
r2
and
b + λn
Dn =
+
a
N −1+c −a
2a
2
=D+
λn
.
a
The following proposition is crucial to prove the lower bounds
Proposition
P
(n)
The series n≥0 pn (t, r , ρ)Zω (η) converges uniformly on compact sets to
p(t, x, y ). Moreover, there exists δ > 0 such that if
|x|
√
t
≤ δ and
X
1
(n)
pn (t, r , ρ)Zω (η) ≤ p0 (t, r , ρ).
n≥1
2
|x|
√
t
≤δ
Lower estimate
Theorem (Metafune, Negro, Spina)
Let p be the heat kernel of L with respect to the measure
dµ(y ) = |y |γ dy . Then
p(t, x, y ) ≥ t
− N2
− γ2
|x|
|y |
− γ2
|x|
1
t2
− N +1+√D
2
c|x−y |2
|y |
−
t
.
∧1
∧
1
e
1
t2
Remark
2 tL
= Ms−1 e tL Ms , where Ms f (x) := f (sx)
x y
− N2 − γ2
p(t, x, y ) = t
p 1, √ , √
t
t
By the scaling property e s
and it is sufficient to prove the statement for t = 1.
Idea of the proof.
Let first
|x|
√
t
|y |
≤ δ, √
≤ δ. Since
t
p(t, x, y ) = p0 (t, r , ρ) +
X
pn (t, r , ρ)Z(n)
ω (η)
n≥1
and
we have
X
1
(n)
≤ p0 (t, r , ρ)
p
(t,
r
,
ρ)Z
(η)
n
ω
2
n≥1
√
1
p(t, x, y ) ≥ p0 (t, r , ρ) ≥ Ct −1− D |r |−s1 |ρ|−s1 ,
2
by the 1d-estimate.
If
|x|
√
t
|y |
≥ δ, √
≥ δ the lower estimate
t
N
p(t, x, y ) ≥ Ct − 2 |y |−γ e −
m|x−y |2
t
follows from
[1] S. Cho, P. Kim, H. Park: Two-sided estimates on Dirichlet heat
kernels for time-dependent operators with singular drifts in C 1,α
domains, Journal of Differential Equations, 252 (2012).
In fact, if we consider L0 := L + |x|b 2 in Cb RN \ B δ with Dirichlet b. c.,
2
then e tL0 is represented by a kernel
q0 (t, x, y ) ' C
|x| −
1∧ √
t
δ
2
!
|y | −
1∧ √
t
δ
2
!
N
t − 2 e −c
In particular for t = 1, |x| ≥ δ, |y | ≥ δ,
2
Ce −c|x−y | ≤ q0 (1, x, y ) ≤ p(1, x, y )|y |γ .
|x−y |2
t
.
Finally, we have to prove the lower bound for t = 1 and |x| ≤ δ ≤ |y |. We
reacall that
2
rρ N−1+c−a X
1
r + ρ2
p(t, x, y ) =
exp −
(r ρ)− 2a
I√Dn
Z(n)
ω (η).
2at
4at
2at
n≥0
and that the lower bound holds when |x| = |y |, or r = ρ, by the
preceeding cases. This gives a lower bound for the sum of the above series
with respect to the variable ξ = r ρ.
Entering this lower bound in the above equality for the kernel p one
concludes the proof.
Green function estimates
We assume λ, D ≥ 0 with D > 0 when λ = 0. The Green function, that is
the integral kernel of (λ − L)−1 , is given by
Z ∞
Gλ (x, y ) :=
e −λt p(t, x, y ) dt, x, y ∈ RN \ {0} .
0
Using the sharp heat kernel estimates we obtain
Z ∞
α
N +1+√D
β2
2
− γ2
−λt − N2
e
t
∧1
exp −
dt
Gλ (x, y ) ' (|x||y |)
t
t
0
where α = |x||y | and β = |x − y | (the reference measure is always
dµ(y ) = |y |γ dy ).
Schrödinger operator
For N > 2, a = 1, c = 0 then γ = 0, D = b +
L=∆−
b
|x|2
N−2 2
2
≥0
is the Schrödinger operator with inverse square potential.
(i) if D > 0, λ ≥ 0,
Gλ (x, y ) ' e −c
√
√D− N−2
2
|x||y
|
λ|x−y |
|x − y |2−N 1 ∧
.
2
|x − y |
(ii) If D = 0, and λ > 0,

2−N
√

e −c λ|x−y | (|x||y |) ∧ |x − y | 2
Gλ (x, y ) '

|x − y |2−N ∨ (|x||y |) 2−N
2 (1 − log(|x − y |)
, |x − y | ≥ 1;
, |x − y | < 1.