本文将Federer的余面积公式推广到属于Sobolev类$W^{1,p}(R^n;R^m)$，$1 \le m < n$，$p>m$的映射$f$，更一般地推广到Lorentz空间$L^{m,1}(R^n)$中具有梯度的映射.这是通过证明$f$在$R^{n+m}$中的图是Hausdorff $n$ -可求积集来实现的。
英文摘要：
We extend Federer's coarea formula to mappings $f$ belonging to the Sobolev class $W^{1,p}(R^n;R^m)$, $1 \le m < n$, $p>m$, and more generally, to mappings with gradient in the Lorentz space $L^{m,1}(R^n)$. This is accomplished by showing that the graph of $f$ in $R^{n+m}$ is a Hausdorff $n$-rectifiable set.
We investigate how the integrability of the derivatives of Orlicz-Sobolev mappings defined on open subsets of $\mathbb{R}^n$ affect the sizes of the images of sets of Hausdorff dimension less than $n$. We measure the sizes of the image sets in terms of generalized Hausdorff measures.
We prove that if $M$ and $N$ are Riemannian, oriented $n$-dimensional manifolds without boundary and additionally $N$ is compact, then Sobolev mappings $W^{1,n}(M,N)$ of finite distortion are continuous. In particular, $W^{1,n}(M,N)$ mappings with almost everywhere positive Jacobian are continuous. This result has been known since 1976 in the case of mappings $W^{1,n}(Ω,\mathbb{R}^n)$, where $Ω\subset\mathbb{R}^n$ is an open set. The case of mappings between manifolds is much more difficult.
We prove that for mappings $W^{1,n}(B^n, \R^n),$ continuous up to the boundary, with modulus of continuity satisfying certain divergence condition, the image of the boundary of the unit ball has zero $n$-Hausdorff measure. For Hölder continuous mappings we also prove an essentially sharp generalized Hausdorff dimension estimate.
In this paper we give characterizations of mappings generate embeddings of Sobolev spaces in the terms of ring capacity inequalities. In addition we prove that such mappings are Lipschitz mappings in the sub-hyperbolic type capacitory metrics.
In the paper we investigate continuity of Orlicz-Sobolev mappings $W^{1,P}(M,N)$ of finite distortion between smooth Riemannian $n$-manifolds, $n\geq 2$, under the assumption that the Young function $P$ satisfies the so called divergence condition $\int_1^\infty P(t)/t^{n+1}\, dt=\infty$. We prove that if the manifolds are oriented, $N$ is compact, and the universal cover of $N$ is not a rational homology sphere, then such mappings are continuous. That includes mappings with $Df\in L^n$ and, more generally, mappings with $Df\in L^n\log^{-1}L$. On the other hand, if the space $W^{1,P}$ is large…
In this paper, we solve the Dirichlet problem for Orlicz-Sobolev maps between singular metric spaces that extends the corresponding result of Guo et al. [arXiv 2021]. As an intermediate step, we develop a version of Rellich-Kondrachov compactness theorem for Orlicz-Sobolev mappings between metric spaces that extends a previous result of Guo and Wenger [Comm. Anal. Geom. 2020]. Another crucial ingredient is an Orlicz-Sobolev extension of the trace theory for metric valued Sobolev maps developed by Korevaar and Schoen [Comm. Anal. Geom. 1993].
The Sard theorem from 1942 requires that a mapping $f:\mathbb{R}^n \to \mathbb{R}^m$ is of class $C^k$, $k > \max (n-m,0)$. In 1957 Duvovitski\uı generalized Sard's theorem to the case of $C^k$ mappings for all $k$. Namely he proved that, for almost all $y\in \mathbb{R}^m$, $\mathcal{H}^{\ell}(C_f \cap f^{-1}(y))=0$ where $\ell = \max(n-m-k+1,0)$, ${\mathcal H}^{\ell}$ denotes the Hausdorff measure, and $C_f$ is the set of critical points of $f$. In 2001 De Pascale proved that the Sard theorem holds true for Sobolev mappings of the class $W_{\rm loc}^{k,p}(\mathbb{R}^n,\mathbb{R}^m)$, $k>\max(…
Universite Claude Bernard Lyon 1（里昂第一大学）；CNRS（法国国家科学研究中心）；Centrale Lyon（里昂中央理工学院）；INSA Lyon（里昂国立应用科学学院）；Université Jean Monnet（让·穆兰大学）；ETH Zürich（苏黎世联邦理工学院）；Université catholique de Louvain, Institut de Recherche en Mathématique et Physique（天主教鲁汶大学数学与物理研究所）
Consider a Lipschitz domain $Ω$ and a measurable function $μ$ supported in $\overlineΩ$ with $\left\|μ\right\|_{L^\infty}<1$. Then the derivatives of a quasiconformal solution of the Beltrami equation $\overline{\partial} f =μ\partial f$ inherit the Sobolev regularity $W^{n,p}(Ω)$ of the Beltrami coefficient $μ$ as long as $Ω$ is regular enough. The condition obtained is that the outward unit normal vector $N$ of the boundary of the domain is in the trace space, that is, $N\in B^{n-1/p}_{p,p}(\partialΩ)$.
Let $ϕ$ be a quasiconformal mapping, and let $T_ϕ$ be the composition operator which maps $f$ to $f\circϕ$. Since $ϕ$ may not be bi-Lipschitz, the composition operator need not map Sobolev spaces to themselves. The study begins with the behavior of $T_ϕ$ on $L^p$ and $W^{1,p}$ for $1<p<\infty$. This cases are well understood but alternative proofs of some known results are provided. Using interpolation techniques it is seen that compactly supported Bessel potential functions in $H^{s,p}$ are sent to $H^{s,q}$ whenever $0<s<1$ for appropriate values of $q$. The techniques used lead to sharp res…
In this paper, we establish Luzin's condition (N) for mappings in certain Sobolev-Orlicz spaces with certain moduli of continuity. Further, given a mapping in these Sobolev-Orlicz spaces, we give bounds on the size of the exceptional set where Luzin's condition (N) may fail. If a mapping violates Luzin's condition (N), we show that there is a Cantor set of measure zero that is mapped to a set of positive measure.
We establish continuity mapping properties of the non-centered fractional maximal operator $M_β$ in the endpoint input space $W^{1,1}(\mathbb{R}^d)$ for $d \geq 2$ in the cases for which its boundedness is known. More precisely, we prove that for $q=d/(d-β)$ the map $f \mapsto |\nabla M_βf|$ is continuous from $W^{1,1}(\mathbb{R}^d)$ to $L^{q}(\mathbb{R}^d)$ for $ 0 < β< 1$ if $f$ is radial and for $1 \leq β< d$ for general $f$. The results for $1\leq β< d$ extend to the centered counterpart $M_β^c$. Moreover, if $d=1$, we show that the conjectured boundedness of that map for $M_β^c$ implies i…
In this paper, we study the mapping properties of the classical Riesz potentials acting on $L^p$-spaces. In the supercritical exponent, we obtain new "almost" Lipschitz continuity estimates for these and related potentials (including, for instance, the logarithmic potential). Applications of these continuity estimates include the deduction of new regularity estimates for distributional solutions to Poisson's equation, as well as a proof of the supercritical Sobolev embedding theorem first shown by Brezis and Wainger in 1980.
