Flat chains in banach spaces

Rectifiability of flat chains in Banach spaces with coefficients in $Z_p$

Aim of this paper is a finer analysis of the group of flat chains with coefficients in $Z_p$ introduced in a recent paper by Ambrosio-Katz, by taking quotients of the group of integer rectifiable currents, along the lines of the the Ziemer and Federer approach. We investigate the typical questions of the theory of currents, namely rectifiability of the measure-theoretic support and boundary rectifiability. Our main result can also be interpreted as a closure theorem for the class of integer rectifiable currents with respect to a (much) weaker convergence, induced by flat distance mod $p$, and …

On the Boundedness of The Bilinear Hilbert Transform along "non-flat" smooth curves. The Banach triangle case ($L^r,\: 1\leq r<\infty$)

We show that the bilinear Hilbert transform $H_Γ$ along curves $Γ=(t,-γ(t))$ with $γ\in\mathcal{N}\mathcal{F}^{C}$ is bounded from $L^{p}(\mathbb{R})\times L^{q}(\mathbb{R})\,\rightarrow\,L^{r}(\mathbb{R})$ where $p,\,q,\,r$ are Hölder indices, i.e. $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$, with $1<p<\infty$, $1<q\leq\infty$ and $1\leq r<\infty$. Here $\mathcal{N}\mathcal{F}^{C}$ stands for a wide class of smooth "non-flat" curves near zero and infinity whose precise definition is given in Section 2. This continues author's earlier work on this topic, extending the boundedness range of $H_Γ$ to a…

Subsets of rectifiable curves in Banach spaces II: universal estimates for almost flat arcs

We prove that in any Banach space the set of windows in which a rectifiable curve resembles two or more straight line segments is quantitatively small with constants that are independent of the curve, the dimension of the space, and the choice of norm. Together with Part I, this completes the proof of the necessary half of the Analyst's Traveling Salesman theorem with sharp exponent in uniformly convex spaces.

The weak convergence of varifolds generated by rectifiable flat G-chains

In the present paper, we prove that the convergence of rectifiable chains in flat norm implies the weak convergence of associated rectifiable varifolds if the limit flat chain is rectifiable and the mass converges also to the mass of limit chain.

Rectifiability of flat chains

We prove (without using Federer's structure theorem) that a finite-mass flat chain over any coefficient group is rectifiable if and only if almost all of its 0-dimensional slices are rectifiable. This implies that every flat chain of finite mass and finite size is rectifiable. It also leads to a simple necessary and sufficient condition on the coefficient group in order for every finite-mass flat chain to be rectifiable.

关于有限质量平坦链的结构

On the structure of flat chains with finite mass

本文证明有限质量平坦链可表示为正规流形在博雷尔集上的限制，并给出实平坦链的逐点刻画及余维一流分解；对群同态诱导的映射，在特定维度上证明满射性。