Recent zbMATH articles in MSC 30Lhttps://www.zbmath.org/atom/cc/30L2022-05-16T20:40:13.078697ZWerkzeugSphericalization and flattening preserve uniform domains in nonlocally compact metric spaceshttps://www.zbmath.org/1483.300562022-05-16T20:40:13.078697Z"Li, Yaxiang"https://www.zbmath.org/authors/?q=ai:li.yaxiang"Ponnusamy, Saminathan"https://www.zbmath.org/authors/?q=ai:ponnusamy.saminathan"Zhou, Qingshan"https://www.zbmath.org/authors/?q=ai:zhou.qingshanSummary: The main aim of this paper is to investigate the invariant properties of uniform domains under flattening and sphericalization in nonlocally compact complete metric spaces. Moreover, we show that quasi-Möbius maps preserve uniform domains in nonlocally compact spaces as well.Weak quasicircles have Lipschitz dimension 1https://www.zbmath.org/1483.301012022-05-16T20:40:13.078697Z"Freeman, David M."https://www.zbmath.org/authors/?q=ai:freeman.david-mandellSummary: We prove that the Lipschitz dimension of any bounded turning Jordan circle or arc is equal to 1. Equivalently, the Lipschitz dimension of any weak quasicircle or arc is equal to 1.Quasiconformal Jordan domainshttps://www.zbmath.org/1483.301022022-05-16T20:40:13.078697Z"Ikonen, Toni"https://www.zbmath.org/authors/?q=ai:ikonen.toniSummary: We extend the classical Carathéodory extension theorem to quasiconformal Jordan domains \((Y, d_Y)\). We say that a metric space \((Y, d_Y)\) is a \textit{quasiconformal Jordan domain} if the completion \(\overline{Y}\) of \((Y, d_Y)\) has finite Hausdorff 2-measure, the \textit{boundary} \(\partial Y = \overline{Y}\setminus Y\) is homeomorphic to \(\mathbb{S}^1\), and there exists a homeomorphism \(\phi : \mathbb{D} \rightarrow (Y, d_Y)\) that is quasiconformal in the geometric sense.
We show that \(\phi\) has a continuous, monotone, and surjective extension \(\Phi : \overline{\mathbb{D}} \rightarrow \overline{Y}\). This result is best possible in this generality. In addition, we find a necessary and sufficient condition for \(\Phi\) to be a quasiconformal homeomorphism. We provide sufficient conditions for the restriction of \(\Phi\) to \(\mathbb{S}^1\) being a quasisymmetry and to \(\partial Y\) being bi-Lipschitz equivalent to a quasicircle in the plane.Uniformization of metric surfaces using isothermal coordinateshttps://www.zbmath.org/1483.301032022-05-16T20:40:13.078697Z"Ikonen, Toni"https://www.zbmath.org/authors/?q=ai:ikonen.toniSummary: We establish a uniformization result for metric surfaces -- metric spaces that are topological surfaces with locally finite Hausdorff \(2\)-measure. Using the geometric definition of quasiconformality, we show that a metric surface that can be covered by quasiconformal images of Euclidean domains is quasiconformally equivalent to a Riemannian surface. To prove this, we construct an atlas of suitable isothermal coordinates.Quasisymmetrically minimal Moran sets on packing dimensionhttps://www.zbmath.org/1483.301042022-05-16T20:40:13.078697Z"Li, Yanzhe"https://www.zbmath.org/authors/?q=ai:li.yanzhe"Fu, Xiaohui"https://www.zbmath.org/authors/?q=ai:fu.xiaohui"Yang, Jiaojiao"https://www.zbmath.org/authors/?q=ai:yang.jiaojiaoQuasisymmetric Koebe uniformization with weak metric doubling measureshttps://www.zbmath.org/1483.301052022-05-16T20:40:13.078697Z"Rajala, Kai"https://www.zbmath.org/authors/?q=ai:rajala.kai"Rasimus, Martti"https://www.zbmath.org/authors/?q=ai:rasimus.marttiSummary: We give a characterization of metric spaces quasisymmetrically equivalent to a finitely connected circle domain. This result generalizes the uniformization of Ahlfors 2-regular spaces by \textit{S. Merenkov} and \textit{K. Wildrick} [Rev. Mat. Iberoam. 29, No. 3, 859--910 (2013; Zbl 1294.30043)].Semiconcavity and sensitivity analysis in mean-field optimal control and applicationshttps://www.zbmath.org/1483.301062022-05-16T20:40:13.078697Z"Bonnet, Benoît"https://www.zbmath.org/authors/?q=ai:bonnet.benoit"Frankowska, Hélène"https://www.zbmath.org/authors/?q=ai:frankowska.heleneSummary: In this article, we investigate some of the fine properties of the value function associated with an optimal control problem in the Wasserstein space of probability measures. Building on new interpolation and linearisation formulas for non-local flows, we prove semiconcavity estimates for the value function, and establish several variants of the so-called sensitivity relations which provide connections between its superdifferential and the adjoint curves stemming from the maximum principle. We subsequently make use of these results to study the propagation of regularity for the value function along optimal trajectories, as well as to investigate sufficient optimality conditions and optimal feedbacks for mean-field optimal control problems.Recent progress in bilinear decompositionshttps://www.zbmath.org/1483.420132022-05-16T20:40:13.078697Z"Fu, Xing"https://www.zbmath.org/authors/?q=ai:fu.xing"Chang, Der-Chen"https://www.zbmath.org/authors/?q=ai:chang.der-chen-e"Yang, Dachun"https://www.zbmath.org/authors/?q=ai:yang.dachunSummary: The targets of this article are twofold. The first one is to give a survey on bilinear decompositions for products of functions in Hardy spaces and their dual spaces, as well as their variants associated with the Schrödinger operator on Euclidean spaces. The second one is to give a new proof of the bilinear decomposition for products of functions in the Hardy space \(H^1\) and BMO on metric measure spaces of homogeneous type. Some applications to div-curl lemmas and commutators are also presented.Rough traces of \textit{BV} functions in metric measure spaceshttps://www.zbmath.org/1483.460352022-05-16T20:40:13.078697Z"Buffa, Vito"https://www.zbmath.org/authors/?q=ai:buffa.vito"Miranda, Michele jun."https://www.zbmath.org/authors/?q=ai:miranda.michele-junSummary: Following a Maz'ya-type approach, we adapt the theory of rough traces of functions of bounded variation (\textit{BV}) to the context of doubling metric measure spaces supporting a Poincaré inequality. This eventually allows for an integration by parts formula involving the rough trace of such functions. We then compare our analysis with the study done in a recent work by
\textit{P.~Lahti} and \textit{N.~Shanmugalingam} [J. Funct. Anal. 274, No.~10, 2754--2791 (2018; Zbl 1392.26016)],
where traces of \textit{BV} functions are studied by means of the more classical Lebesgue-point characterization, and we determine the conditions under which the two notions coincide.Orlicz-Sobolev inequalities and the doubling conditionhttps://www.zbmath.org/1483.460362022-05-16T20:40:13.078697Z"Korobenko, Lyudmila"https://www.zbmath.org/authors/?q=ai:korobenko.lyudmilaSummary: In [\textit{L. Korobenko} et al., Proc. Am. Math. Soc. 143, No.~9, 4017--4028 (2015; Zbl 1325.35055)] it has been shown that a \((p,q)\) Sobolev inequality with \(p>q\) implies the doubling condition on the underlying measure. We show that even weaker Orlicz-Sobolev inequalities, where the gain on the left-hand side is smaller than any power bump, imply doubling. Moreover, we derive a condition on the quantity that should replace the radius on the righ-hand side (which we call ``superradius''), that is necessary to ensure that the space can support the Orlicz-Sobolev inequality and simultaneously be non-doubling.