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Abstract
Let
A (
⋅ )
be an
( n + 1 ) × ( n + 1 )
uniformly elliptic matrix with Hölder continuous real coefficients and let
ℰ A ( x , y ) be the fundamental
solution of the PDE
div A (
⋅ ) ∇ u
= 0
in
ℝ n + 1 . Let
μ be a compactly
supported
n -AD-regular
measure in
ℝ n + 1
and consider the associated operator
T μ f ( x )
= ∫
∇ x ℰ A ( x , y ) f ( y ) d μ ( y ) .
We show that if
T μ
is bounded in
L 2 ( μ ) ,
then
μ is uniformly
n -rectifiable. This extends the
solution of the codimension-1
David–Semmes problem for the Riesz transform to the gradient of the single-layer potential.
Together with a previous result of Conde-Alonso, Mourgoglou and Tolsa, this shows that, given
E
⊂ ℝ n + 1 with finite
Hausdorff measure
ℋ n ,
if
T ℋ n | E is
bounded in
L 2 ( ℋ n | E ) ,
then
E is
n -rectifiable.
Further, as an application we show that if the elliptic measure associated to
the above PDE is absolutely continuous with respect to surface measure,
then it must be rectifiable, analogously to what happens with harmonic
measure.
Keywords
David–Semmes problem, uniform rectifiability, gradient of
the single-layer potential, elliptic measure,
rectifiability
Mathematical Subject Classification 2010
Primary: 28A75, 31B15, 35J15, 42B37
Milestones
Received: 26 November 2018
Revised: 5 September 2019
Accepted: 21 November 2019
Published: 18 May 2021