KFP operators with coefficients measurable in time and Dini continuous in space
Authors: Biagi S. , Bramanti M. , Stroffolini B. 
Published: April 2024

We consider degenerate Kolmogorov–Fokker–Planck operators $$\begin{aligned} \mathcal {L}u&=\sum _{i,j=1}^{m_{0}}a_{ij}(x,t)\partial _{x_{i}x_{j}} ^{2}u+\sum _{k,j=1}^{N}b_{jk}x_{k}\partial _{x_{j}}u-\partial _{t}u\\&\equiv \sum _{i,j=1}^{m_{0}}a_{ij}(x,t)\partial _{x_{i}x_{j}}^{2}u+Yu \end{aligned}$$

L
u

=

∑

i
,
j
=
1

m
0

a

ij

(
x
,
t
)

∂

x
i

x
j

2

u
+

∑

k
,
j
=
1

N

b

jk

x
k

∂

x
j

u
-

∂
t

u

≡

∑

i
,
j
=
1

m
0

a

ij

(
x
,
t
)

∂

x
i

x
j

2

u
+
Y
u

(with $$(x,t)\in \mathbb {R}^{N+1}$$

(
x
,
t
)

∈

R

N
+
1

and $$1\le m_{0}\le N$$

1
≤

m
0

≤
N

) such that the corresponding model operator having constant $$a_{ij}$$

a

ij

is hypoelliptic, translation invariant w.r.t. a Lie group operation in $$\mathbb {R} ^{N+1}$$

R

N
+
1

and 2-homogeneous w.r.t. a family of nonisotropic dilations. The matrix $$(a_{ij})_{i,j=1}^{m_{0}}$$

(

a

ij

)

i
,
j
=
1

m
0

is symmetric and uniformly positive on $$\mathbb {R}^{m_{0}}$$

R

m
0

. The coefficients $$a_{ij}$$

a

ij

are bounded and Dini continuous in space, and only bounded measurable in time. This means that, setting $$\begin{aligned} \mathrm {(i)}&\,\,S_{T}=\mathbb {R}^{N}\times \left( -\infty ,T\right) ,\\ \mathrm {(ii)}&\,\,\omega _{f,S_{T}}(r) = \sup _{\begin{array}{c} (x,t),(y,t)\in S_{T}\\ \Vert x-y\Vert \le r \end{array}}\vert f(x,t) -f(y,t)\vert \\ \mathrm {(iii)}&\,\,\Vert f\Vert _{\mathcal {D}( S_{T}) } =\int _{0}^{1} \frac{\omega _{f,S_{T}}(r) }{r}dr+\Vert f\Vert _{L^{\infty }\left( S_{T}\right) } \end{aligned}$$

(
i
)

S
T

=

R

N

×

-
∞
,
T

,

(
ii
)

ω

f
,

S
T

(
r
)

=

sup

(
x
,
t
)

,

(
y
,
t
)

∈

S
T

‖
x
-
y
‖
≤
r

|
f

(
x
,
t
)

-
f

(
y
,
t
)

|

(
iii
)

‖
f
‖

D
(

S
T

)

=

∫

0

1

ω

f
,

S
T

(
r
)

r

d
r
+

‖
f
‖

L
∞

S
T

we require the finiteness of $$\Vert a_{ij}\Vert _{\mathcal {D}(S_{T})}$$

‖

a

ij

‖

D
(

S
T

)

. We bound $$\omega _{u_{x_{i}x_{j}},S_{T}}$$

ω

u

x
i

x
j

,

S
T

, $$\Vert u_{x_{i}x_{j}}\Vert _{L^{\infty }( S_{T}) }$$

‖

u

x
i

x
j

‖

L
∞

(

S
T

)

($$i,j=1,2,...,m_{0}$$

i
,
j
=
1
,
2
,
.
.
.
,

m
0

), $$\omega _{Yu,S_{T}}$$

ω

Y
u
,

S
T

, $$\Vert Yu\Vert _{L^{\infty }( S_{T}) }$$

‖
Y
u
‖

L
∞

(

S
T

)

in terms of $$\omega _{\mathcal {L}u,S_{T}}$$

ω

L
u
,

S
T

, $$\Vert \mathcal {L}u\Vert _{L^{\infty }( S_{T}) }$$

‖
L
u
‖

L
∞

(

S
T

)

and $$\Vert u\Vert _{L^{\infty }\left( S_{T}\right) }$$

‖
u
‖

L
∞

S
T

, getting a control on the uniform continuity in space of $$u_{x_{i}x_{j}},Yu$$

u

x
i

x
j

,
Y
u

if $$\mathcal {L}u$$

L
u

is bounded and Dini-continuous in space. Under the additional assumption that both the coefficients $$a_{ij}$$

a

ij

and $$\mathcal {L}u$$

L
u

are log-Dini continuous, meaning the finiteness of the quantity $$\begin{aligned} \int _{0}^{1}\frac{\omega _{f,S_{T}}\left( r\right) }{r}\left| \log r\right| dr, \end{aligned}$$

∫

0

1

ω

f
,

S
T

r

r

log
r

d
r
,

we prove that $$u_{x_{i}x_{j}}$$

u

x
i

x
j

and Yu are Dini continuous; moreover, in this case, the derivatives $$u_{x_{i}x_{j}}$$

u

x
i

x
j

are locally uniformly continuous in space and time.