109 research outputs found
Semi-uniform domains and the property for harmonic measure
We study the properties of harmonic measure in semi-uniform domains. Aikawa
and Hirata showed in \cite{AH08} that, for John domains satisfying the capacity
density condition (CDC), the doubling property for harmonic measure is
equivalent to the domain being semi-uniform. Our first result removes the John
condition by showing that any domain satisfying the CDC whose harmonic measure
is doubling is semi-uniform. Next, we develop a substitute for some classical
estimates on harmonic measure in nontangentially accessible domains that works
in semi-uniform domains.
We also show that semi-uniform domains with uniformly rectifiable boundary
have big pieces of chord-arc subdomains. We cannot hope for big pieces of
Lipschitz subdomains (as was shown for chord-arc domains by David and Jerison
\cite{DJ90}) due to an example of Hrycak, which we review in the appendix.
Finally, we combine these tools to study the -property of
harmonic measure. For a domain with Ahlfors-David regular boundary, it was
shown by Hofmann and Martell that the property of harmonic measure
implies uniform rectifiability of the boundary \cite{HM15,HLMN17} . Since
-weights are doubling, this also implies the domain is
semi-uniform. Our final result shows that these two properties, semi-uniformity
and uniformly rectifiable boundary, also imply the property for
harmonic measure, thus classifying geometrically all domains for which this
holds.Comment: Corrected some typos/errors, the statement of Theorem II and the
harnack chain definition of uniform domains, added a referenc
Dimension drop for harmonic measure on Ahlfors regular boundaries
We show that given a domain with uniformly
non-flat Ahlfors -regular boundary and , the dimension of its
harmonic measure is strictly less than .Comment: Corrected typos and mistakes (in particular, a part of Lemma 4.1 was
poorly written), made minor corrections to statements of Lemma 4.1 and Lemma
4.2, amended introduction slightly, added a figure. To appear in Potential
Analysi
Bi-Lipschitz parts of quasisymmetric mappings
A natural quantity that measures how well a map is approximated by an affine transformation is
where the infimum ranges over all non constant affine transformations. This is
natural insofar as it is invariant under rescaling in either its domain or
image. We show that if is
quasisymmetric and its image has a sufficient amount of rectifiable structure
(although not necessarily -finite), then
is a Carleson measure on
. Moreover, this is an equivalence: the
existence of such a Carleson measure implies that, in every ball
, there is a set occupying 90 of
, say, upon which is bi-Lipschitz (and hence guaranteeing
rectifiable pieces in the image).
En route, we make a minor adjustment to a theorem of Semmes to show that
quasisymmetric maps of subsets of into are
bi-Lipschitz on a large subset quantitatively.Comment: Corrected several proofs, reorganized introduction, added a
references, changed title (previously "Quantitative differentiation of
quasisymmetric mappings in Euclidean space". Accepted to Revist
Tangents, rectifiability, and corkscrew domains
In a recent paper, Cs\"ornyei and Wilson prove that curves in Euclidean space
of -finite length have tangents on a set of positive
-measure. They also show that a higher dimensional analogue of
this result is not possible without some additional assumptions. In this note,
we show that if has the property that each
ball centered on contains two large balls in different components of
and has -finite -measure, then
it has -dimensional tangent points in a set of positive
-measure. We also give shorter proofs that Semmes surfaces are
uniformly rectifiable and, if is an exterior
corkscrew domain whose boundary has locally finite -measure,
one can find a Lipschitz subdomain intersecting a large portion of the
boundary.Comment: Corrected proofs and typos, changed a definition. To appear in
Publicacions Matem\`atique
Tangent measures and absolute continuity of harmonic measure
We show that for uniform domains whose
boundaries satisfy a certain nondegeneracy condition that harmonic measure
cannot be mutually absolutely continuous with respect to -dimensional
Hausdorff measure unless . We employ a lemma that shows that at
almost every nondegenerate point, we may find a tangent measure of harmonic
measure whose support is the boundary of yet another uniform domain whose
harmonic measure resembles the tangent measure.Comment: Added a figure, corrected a few typos, added a reference. To appear
in Revista Matem\'atica Iberoamerican
A characterization of -rectifiable doubling measures with connected supports
Garnett, Killip, and Schul have exhibited a doubling measure with
support equal to which is -rectifiable, meaning there are
countably many curves of finite length for which
. In this note, we
characterize when a doubling measure with support equal to a connected
metric space has a -rectifiable subset of positive measure and show this
set coincides up to a set of -measure zero with the set of for
which .Comment: Made suggested referee changes. Published in Anal. PDE 9 (2016), no.
1, 99--10
Quantitative Comparisons of Multiscale Geometric Properties
We generalize some characterizations of uniformly rectifiable (UR) sets to
sets whose Hausdorff content is lower regular (and in particular, do not need
to be Ahlfors regular). For example, David and Semmes showed that, given an
Ahlfors -regular set , if we consider the set of surface
cubes (in the sense of Christ and David) near which does not look
approximately like a union of planes, then is UR if and only if
satisfies a Carleson packing condition, that is, for any surface
cube , We show that, for lower content regular sets
that aren't necessarily Ahlfors regular, if denotes the square
sum of -numbers over subcubes of as in the Traveling Salesman
Theorem for higher dimensional sets [AS18], then We prove similar results for other uniform
rectifiability critera, such as the Local Symmetry, Local Convexity, and
Generalized Weak Exterior Convexity conditions.
En route, we show how to construct a corona decomposition of any lower
content regular set by Ahlfors regular sets, similar to the classical corona
decomposition of UR sets by Lipschitz graphs developed by David and Semmes.Comment: 39 pages. To appear in Analysis & PDE
How to take shortcuts in Euclidean space: making a given set into a short quasi-convex set
For a given connected set in dimensional Euclidean space, we
construct a connected set such that the two sets
have comparable Hausdorff length, and the set has the property
that it is quasiconvex, i.e. any two points and in can
be connected via a path, all of which is in , which has length
bounded by a fixed constant multiple of the Euclidean distance between and
. Thus, for any set in dimensional Euclidean space we have a set
as above such that has comparable Hausdorff
length to a shortest connected set containing . Constants appearing here
depend only on the ambient dimension . In the case where is
Reifenberg flat, our constants are also independent the dimension , and in
this case, our theorem holds for in an infinite dimensional Hilbert
space. This work closely related to spanners, which appear in computer
science. Keywords: chord-arc, quasiconvex, k-spanner, traveling salesman.Comment: Made referee edit
Mutual absolute continuity of interior and exterior harmonic measure implies rectifiability
We show that, for disjoint domains in the Euclidean space whose boundaries
satisfy a non-degeneracy condition, mutual absolute continuity of their
harmonic measures implies absolute continuity with respect to surface measure
and rectifiability in the intersection of their boundaries.Comment: Made suggested changes by referee. To appear in Comm. Pure Appl. Mat
Absolute continuity of harmonic measure for domains with lower regular boundaries
We study absolute continuity of harmonic measure with respect to surface
measure on domains that have large complements. We show that if
is -Ahlfors regular and splits into two NTA domains then on . This result is a natural
generalisation of a result of Wu in [Wu86].
We also prove that almost every point in is a cone
point if is a Lipschitz graph. Combining these results and a result
from [AHMMMTV], we characterize sets of absolute continuity with finite
-measure both in terms of the cone point condition and in
terms of the rectifiable structure of the boundary. This generalizes the
results of McMillan in [McM69] and Pommerenke in [Pom86].
Finally, we also show our first result holds for elliptic measure associated
with real second order divergence form elliptic operators with a mild
assumption on the gradient of the matrix.Comment: Corrected several typos and error
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