arXiv subjects
Publications and source records attributed to Jonas Azzam.
For a compact connected set $X\subseteq \ell^{\infty}$, we define a quantity $β'(x,r)$ that measures how close $X$ may be approximated in a ball $B(x,r)$ by a geodesic curve. We then show there is $c>0$ so that if $β'(x,r)>β>0$ for all $x\in X$ and $r 1+cβ^{2}$. This generalizes a theorem of Bishop and Jones and answers a question posed by Bishop and Tyson.
We consider a number of uniqueness questions for several wide classes of active scalar equations, unifying and generalizing the techniques of several authors. As special cases of our results, we provide a significantly simplified proof to the known uniqueness result for the 2D Euler equations in $L^1 \cap BMO$ and provide a mild improvement to the recent results of Rusin for the 2D inviscid surface quasi-geostrophic (SQG) equations, which are now to our knowledge, the best results known for this model. We also obtain what are (to our knowledge) the strongest known uniqueness results for the Patlak-Keller-Segel models. We obtain these results via technical refinements of energy methods which are well-known in the $L^2$ setting but are less well-known in the $\dot{H}^{-1}$ setting. The $\dot{H}^{-1}$ method can be considered a generalization of Yudovich's classical method and is naturally applied to equations such as the Patlak-Keller-Segel models with nonlinear diffusion, and other variants. Important points of our analysis are an $L^p$-$BMO$ interpolation lemma and a Sobolev embedding lemma which shows that velocity fields $v$ with $\grad v \in BMO$ are locally log-Lipschitz; the latter is known in harmonic analysis but does not seem to have been connected to this setting.
The purpose of this note is to point out a simple consequence of some earlier work of the authors, "Hard Sard: Quantitative implicit function and extension theorems for Lipschitz maps". For $f$, a Lipschitz function from a Euclidean space into a metric space, we give quantitative estimates for how often the pullback of the metric under $f$ is approximately a seminorm. This is a quantitative version of Kirchheim's metric differentiation result from 1994. Our result is in the form of a Carleson-type estimate.
We prove a global implicit function theorem. In particular we show that any Lipschitz map $f:\bR^n\times \bR^m\to\bR^n$ (with $n$-dim. image) can be precomposed with a bi-Lipschitz map $\bar{g}:\bR^n\times \bR^m\to \bR^n\times \bR^m$ such that $f\circ \bar{g}$ will satisfy, when we restrict to a large portion of the domain $E\subset \bR^n\times \bR^m$, that $f\circ \bar{g}$ is bi-Lipschitz in the first coordinate, and constant in the second coordinate. Geometrically speaking, the map $\bar{g}$ distorts $\bR^{n+m}$ in a controlled manner, so that the fibers of $f$ are straightened out. Furthermore, our results stay valid when the target space is replaced by {\bf any metric space}. A main point is that our results are quantitative: the size of the set $E$ on which behavior is good is a significant part of the discussion. Our estimates are motivated by examples such as Kaufman's 1979 construction of a $C^1$ map from $[0,1]^3$ onto $[0,1]^2$ with rank $\leq 1$ everywhere. On route we prove an extension theorem which is of independent interest. We show that for any $D\geq n$, any Lipschitz function $f:[0,1]^n\to \bR^D$ gives rise to a large (in an appropriate sense) subset $E\subset [0,1]^n$ such that $f|_E$ is bi-Lipschitz and may be extended to a bi-Lipschitz function defined on {\bf all} of $\bR^n$. The most interesting case is the case $D=n$. As a simple corollary, we show that $n$-dimensional Ahlfors-David regular spaces lying in $\bR^{D}$ having big pieces of bi-Lipschitz images also have big pieces of big pieces of Lipschitz graphs in $\bR^{D}$. This was previously known only for $D\geq 2n+1$ by a result of G. David and S. Semmes.
For a given connected set $Γ$ in $d-$dimensional Euclidean space, we construct a connected set $\tildeΓ\supset Γ$ such that the two sets have comparable Hausdorff length, and the set $\tildeΓ$ has the property that it is quasiconvex, i.e. any two points $x$ and $y$ in $\tildeΓ$ can be connected via a path, all of which is in $\tildeΓ$, which has length bounded by a fixed constant multiple of the Euclidean distance between $x$ and $y$. Thus, for any set $K$ in $d-$dimensional Euclidean space we have a set $\tildeΓ$ as above such that $\tildeΓ$ has comparable Hausdorff length to a shortest connected set containing $K$. Constants appearing here depend only on the ambient dimension $d$. In the case where $Γ$ is Reifenberg flat, our constants are also independent the dimension $d$, and in this case, our theorem holds for $Γ$ in an infinite dimensional Hilbert space. This work closely related to $k-$spanners, which appear in computer science. Keywords: chord-arc, quasiconvex, k-spanner, traveling salesman.
We prove that there exists $M>0$ such that for any closed rectifiable curve $Γ$ in Hilbert space, almost every point in $Γ$ is contained in a countable union of $M$ chord-arc curves whose total length is no more than $M$ times the length of $Γ$.