arXiv subjects
Publications and source records attributed to Panu Lahti.
In a complete metric space equipped with a doubling measure and supporting a $(1,1)$-Poincaré inequality, we show that every set satisfying a suitable capacitary density condition is removable for Newton-Sobolev functions.
Given a homeomorphism $f\colon X\to Y$ between $Q$-dimensional spaces $X,Y$, we show that $f$ satisfying the metric definition of quasiconformality outside suitable exceptional sets implies that $f$ belongs to the Sobolev class $N_{\rm{loc}}^{1,p}(X;Y)$, where $1< p\le Q$, and also implies one direction of the geometric definition of quasiconformality. Unlike previous results, we only assume a pointwise version of Ahlfors $Q$-regularity, which particularly enables various weighted spaces to be included in the theory. Unexpectedly, we can apply this to obtain results that are new even in the classical Euclidean setting. In particular, in spaces including the Carnot groups, we are able to prove the Sobolev regularity $f\in N_{\rm{loc}}^{1,Q}(X;Y)$ without the strong assumption of the infinitesimal distortion $h_f$ belonging to $L^{\infty}(X)$.
We study a characterization of BV and Sobolev functions via nonlocal functionals in metric spaces equipped with a doubling measure and supporting a Poincaré inequality. Compared with previous works, we consider more general functionals. We also give a counterexample in the case $p=1$ demonstrating that unlike in Euclidean spaces, in metric measure spaces the limit of the nonlocal functions is only comparable, not necessarily equal, to the variation measure $\| Df\|(Ω)$.
Consider a mapping $f\colon X\to Y$ between two metric measure spaces. We study generalized versions of the local Lipschitz number $\mathrm{Lip} f$, as well as of the distortion number $H_f$ that is used to define quasiconformal mappings. Using these, we give sufficient conditions for $f$ being a BV mapping $f\in BV_{\mathrm{loc}}(X;Y)$ or a Newton-Sobolev mapping $f\in N_{\mathrm{loc}}^{1,p}(X;Y)$, with $1\le p<\infty$.
We study pointwise convergence properties of weakly* converging sequences $\{u_i\}_{i \in {\mathbb N}}$ in $\mathrm{BV}({\mathbb R}^n)$. We show that, after passage to a suitable subsequence (not relabeled), we have pointwise convergence $u_i^*(x)\to u^*(x)$ of the precise representatives for all $x\in {\mathbb R}^n \setminus E$, where the exceptional set $E \subset {\mathbb R}^n$ has on the one hand Hausdorff dimension at most $n-1$, and is on the other hand also negligible with respect to the Cantor part of $|D u|$. Furthermore, we discuss the optimality of these results.
Following Malý's definition of absolutely continuous functions of several variables, we consider $Q$-absolutely continuous mappings $f\colon X\to V$ between a doubling metric measure space $X$ and a Banach space $V$. The relation between these mappings and Sobolev mappings $f\in N^{1,p}(X;V)$ for $p\ge Q$ is investigated. In particular, a locally $Q$-absolutely continuous mapping on an Ahlfors $Q$-regular space is a continuous mapping in $N^{1,Q}_{\rm{loc}}(X;V)$, as well as differentiable almost everywhere in terms of Cheeger derivatives provided $V$ satisfies the Radon-Nikodym property. Conversely, though a continuous Sobolev mapping $f\in N^{1,Q}_{\rm{loc}}(X;V)$ is generally not locally $Q$-absolutely continuous, this implication holds if $f$ is further assumed to be pseudomonotone. It follows that pseudomonotone mappings satisfying a relaxed quasiconformality condition are also $Q$-absolutely continuous.
In metric measure spaces, we study boundary traces of BV functions in domains equipped with a doubling measure and supporting a Poincaré inequality, but possibly having a very large and irregular boundary. We show that the trace exists in the ordinary sense in a certain part of the boundary, and that this part is sufficient to determine the integrability of the rough trace, as well as the possibility of zero extending the function to the whole space as a BV function.
We show that a mapping $f\colon X\to Y$ satisfying the metric condition of quasiconformality outside suitable exceptional sets is in the Newton-Sobolev class $N_{\textrm{loc}}^{1,1}(X;Y)$. Contrary to previous works, we only assume an asymptotic version of Ahlfors-regularity on $X,Y$. This allows many non-Ahlfors regular spaces, such as weighted spaces and Fred Gehring's bowtie, to be included in the theory. Unexpectedly, already in the classical setting of unweighted Euclidean spaces, our theory detects Sobolev mappings that are not recognized by previous results.
Bonicatto--Pasqualetto--Rajala (2020) proved that a decomposition theorem for sets of finite perimeter into indecomposable sets, known to hold in Euclidean spaces, holds also in complete metric spaces equipped with a doubling measure, supporting a Poincaré inequality, and satisfying an \emph{isotropicity} condition. We show that the last assumption can be removed.
We show that the non-centered maximal function of a BV function is quasicontinuous. We also show that \emph{if} the non-centered maximal functions of an SBV function is a BV function, then it is in fact a Sobolev function. Using a recent result of Weigt, we are in particular able to show that the non-centered maximal function of a set of finite perimeter is a Sobolev function.
In this paper, we study the asymptotic behavior of BV functions in complete metric measure spaces equipped with a doubling measure supporting a $1$-Poincaré inequality. We show that at almost every point $x$ outside the Cantor and jump parts of a BV function, the asymptotic limit of the function is a Lipschitz continuous function of least gradient on a tangent space to the metric space based at $x$. We also show that, at co-dimension $1$ Hausdorff measure almost every measure-theoretic boundary point of a set $E$ of finite perimeter, there is an asymptotic limit set $(E)_\infty$ corresponding to the asymptotic expansion of $E$ and that every such asymptotic limit $(E)_\infty$ is a quasiminimal set of finite perimeter. We also show that the perimeter measure of $(E)_\infty$ is Ahlfors co-dimension $1$ regular.
We study the boundary traces of Newton-Sobolev, Hajlasz-Sobolev, and BV (bounded variation) functions. Assuming less regularity of the domain than is usually done in the literature, we show that all of these function classes achieve the same "boundary values", which in particular implies that the trace spaces coincide provided that they exist. Many of our results seem to be new even in Euclidean spaces but we work in a more general complete metric space equipped with a doubling measure and supporting a Poincare inequality.
In this paper, we study functions of bounded variation on a complete and connected metric space with finite one-dimensional Hausdorff measure. The definition of BV functions on a compact interval based on pointwise variation is extended to this general setting. We show this definition of BV functions is equivalent to the BV functions introduced by Miranda. Furthermore, we study the necessity of conditions on the underlying space in Federer's characterization of sets of finite perimeter on metric measure spaces. In particular, our examples show that the doubling and Poincaré inequality conditions are essential in showing that a set has finite perimeter if the codimension one Hausdorff measure of the measure-theoretic boundary is finite.
Federer's characterization states that a set $E\subset \mathbb{R}^n$ is of finite perimeter if and only if $\mathcal H^{n-1}(\partial^*E)<\infty$. Here the measure-theoretic boundary $\partial^*E$ consists of those points where both $E$ and its complement have positive upper density. We show that the characterization remains true if $\partial^*E$ is replaced by a smaller boundary consisting of those points where the \emph{lower} densities of both $E$ and its complement are at least a given number. This result is new even in Euclidean spaces but we prove it in a more general complete metric space that is equipped with a doubling measure and supports a Poincaré inequality.
We prove a duality relation for the moduli of the family of curves connecting two sets and the family of surfaces separating the sets, in the setting of a complete metric space equipped with a doubling measure and supporting a Poincaré inequality. Then we apply this to show that quasiconformal mappings can be characterized by the fact that they quasi-preserve the modulus of certain families of surfaces.
In a complete metric space that is equipped with a doubling measure and supports a Poincaré inequality, we study strict subsets, i.e. sets whose variational capacity with respect to a larger reference set is finite, in the case $p=1$. Relying on the concept of fine topology, we give a characterization of those strict subsets that are also sets of finite perimeter, and then we apply this to the study of condensers as well as BV capacities. We also apply the theory to prove a pointwise approximation result for functions of bounded variation.
We study fine potential theory and in particular partitions of unity in quasiopen sets in the case $p=1$. Using these, we develop an analog of the discrete convolution technique in quasiopen (instead of open) sets. We apply this technique to show that every function of bounded variation (BV function) can be approximated in the BV and $L^{\infty}$ norms by BV functions whose jump sets are of finite Hausdorff measure. Our results seem to be new even in Euclidean spaces but we work in a more general complete metric space that is equipped with a doubling measure and supports a Poincaré inequality.
We prove a Leibniz rule for BV functions in a complete metric space that is equipped with a doubling measure and supports a Poincaré inequality. Unlike in previous versions of the rule, we do not assume the functions to be locally essentially bounded and the end result does not involve a constant $C\ge 1$, and so our result seems to be essentially the best possible. In order to obtain the rule in such generality, we first study the weak* convergence of the variation measure of BV functions, with quasi semicontinuous test functions.