L2-Boundedness of Gradients of Single Layer Potentials for Elliptic Operators with Coefficients of Dini Mean Oscillation-Type
Authors: Molero Alejandro , Mourgoglou Mihalis , Puliatti Carmelo , Tolsa Xavier 
Published: April 2023

We consider a uniformly elliptic operator $$L_A$$

L
A

in divergence form associated with an $$(n+1)\times (n+1)$$

(
n
+
1
)
×
(
n
+
1
)

-matrix A with real, merely bounded, and possibly non-symmetric coefficients. If "Equation missing"then, under suitable Dini-type assumptions on $$\omega _A$$

ω
A

, we prove the following: if $$\mu $$
μ
is a compactly supported Radon measure in $$\mathbb {R}^{n+1}$$

R

n
+
1

, $$n \ge 2$$

n
≥
2

, and $$ T_\mu f(x)=\int \nabla _x\Gamma _A (x,y)f(y)\, \textrm{d}\mu (y) $$

T
μ

f

(
x
)

=
∫

∇
x

Γ
A

(
x
,
y
)

f

(
y
)

d
μ

(
y
)

denotes the gradient of the single layer potential associated with $$L_A$$

L
A

, then $$\begin{aligned} 1+ \Vert T_\mu \Vert _{L^2(\mu )\rightarrow L^2(\mu )}\approx 1+ \Vert {\mathcal {R}}_\mu \Vert _{L^2(\mu )\rightarrow L^2(\mu )}, \end{aligned}$$

1
+
‖

T
μ

‖

L
2

(
μ
)

→

L
2

(
μ
)

≈
1
+

‖

R
μ

‖

L
2

(
μ
)

→

L
2

(
μ
)

,

where $${\mathcal {R}}_\mu $$

R
μ

indicates the n-dimensional Riesz transform. This allows us to provide a direct generalization of some deep geometric results, initially obtained for $${\mathcal {R}}_\mu $$

R
μ

, which were recently extended to $$T_\mu $$

T
μ

associated with $$L_A$$

L
A

with Hölder continuous coefficients. In particular, we show the following:

If $$\mu $$
μ
is an n-Ahlfors-David-regular measure on $$\mathbb {R}^{n+1}$$

R

n
+
1

with compact support, then $$T_\mu $$

T
μ

is bounded on $$L^2(\mu )$$

L
2

(
μ
)

if and only if $$\mu $$
μ
is uniformly n-rectifiable.

Let $$E\subset \mathbb {R}^{n+1}$$

E
⊂

R

n
+
1

be compact and $${\mathcal {H}}^n(E)<\infty $$

H

n

(
E
)

<
∞

. If $$T_{{\mathcal {H}}^n|_E}$$

T

H

n

|

E

is bounded on $$L^2({\mathcal {H}}^n|_E)$$

L
2

(

H

n

|
E

)

, then E is n-rectifiable.

If $$\mu $$
μ
is a non-zero measure on $$\mathbb {R}^{n+1}$$

R

n
+
1

such that $$\limsup _{r\rightarrow 0}\tfrac{\mu (B(x,r))}{(2r)^n}$$

lim sup

r
→
0

μ
(
B
(
x
,
r
)
)

(
2
r
)

n

is positive and finite for $$\mu $$
μ
-a.e. $$x\in \mathbb {R}^{n+1}$$

x
∈

R

n
+
1

and $$\liminf _{r\rightarrow 0}\tfrac{\mu (B(x,r))}{(2r)^n}$$

lim inf

r
→
0

μ
(
B
(
x
,
r
)
)

(
2
r
)

n

vanishes for $$\mu $$
μ
-a.e. $$x\in \mathbb {R}^{n+1}$$

x
∈

R

n
+
1

, then the operator $$T_\mu $$

T
μ

is not bounded on $$L^2(\mu )$$

L
2

(
μ
)

.

Finally, we prove that if $$\mu $$
μ
is a Radon measure on $${\mathbb {R}}^{n+1}$$

R

n
+
1

with compact support which satisfies a proper set of local conditions at the level of a ball $$B=B(x,r)\subset {\mathbb {R}}^{n+1}$$

B
=
B

(
x
,
r
)

⊂

R

n
+
1

such that $$\mu (B)\approx r^n$$

μ

(
B
)

≈

r
n

and r is small enough, then a significant portion of the support of $$\mu |_B$$

μ
|

B

can be covered by a uniformly n-rectifiable set. These assumptions include a flatness condition, the $$L^2(\mu )$$

L
2

(
μ
)

-boundedness of $$T_\mu $$

T
μ

on a large enough dilation of B, and the smallness of the mean oscillation of $$T_\mu $$

T
μ

at the level of B.