Tapio Rajala, Aleksandra Zapadinskaya, Thomas Zürcher
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.
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This paper is served as a first contribution regarding the boundedness of Hausdorff operators on function spaces with smoothness. The sharp conditions are established for boundedness of Hausdorff operators on Sobolev spaces $W^{k,1}$. As applications, some bounded and unbounded properties of Hardy operator and adjoint Hardy operator on $W^{k,1}$ are deduced.
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.
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.
In this paper we prove that if f is a planar K-quasiconformal map and 0<t<2, t' = 2t/(2K-Kt+t), then f transforms sets of finite (t')-Hausdorff measure into sets of finite t-Hausdorff measure. We also prove the following more quantitative statement: If E is a planar set, then H^t(E) \leq C(K) H^{t'}(f(E))^{t/(t'K)}, where H^s stands for the s-Hausdorff measure.
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…
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.
Polynomial Szemerédi for sets with large Hausdorff dimension on the Torus
本文证明了环面上豪斯多夫维数大于1-ε的紧集必包含多项式构型，并由此推出相关多项式多重遍历平均发散集的维数严格小于1。
We prove an estimate on the Hausdorff-dimension of the set of two-sided boundary points of general Sobolev-extension domains on Euclidean spaces. We also present examples showing lower bounds on possible dimension estimates of this type.
Tangents to Lipschitz and Sobolev images
本文研究了Lipschitz和Sobolev映射像的切线性质，证明了在特定条件下图像具有唯一的切集或切线，并展示了如何通过包装内容有限性来升级近切线。
Sharp Poincaré--Sobolev Inequalities of Choquet--Lorentz Integrals with Respect to Hausdorff Contents on Bounded John Domains
本文证明了有界 John 域上关于 Hausdorff 容度的 Choquet--Lorentz 积分具有精确指数范围的 Poincaré--Sobolev 不等式，发展了分数阶极大不等式等新工具，并给出了临界情形的端点估计及 Riesz 位势的精确有界性。
In this paper we give an estimate for the Hausdorff dimension of the set of two-sided points of the boundary of bounded simply connected Sobolev $W^{1,p}$-extension domain for $1<p<2$. Sharpness of the estimate is shown by examples. We also prove the equivalence of different definitions of two-sided points.
Intermediate dimensions of measures: Interpolating between Hausdorff and Minkowski dimensions
本文定义了介于豪斯多夫和闵可夫斯基维度之间的测度中间维度，填补了测度维度插值研究中的空白，并提出了反Frostman引理和容量理论方法。
Limits at infinity for Hajłasz-Sobolev functions in metric spaces
本文研究一致完全度量空间上齐次 Hajłasz-Sobolev 函数在无穷远处的极限行为，利用变分相对容量刻画例外集，改进并推广了基于 Hausdorff 容量的已有结果。
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(…
We establish Luzin N and Morse--Sard properties for functions from the Sobolev space $W^{n,1}({\mathbb R}^{n})$. Using these results we prove that almost all level sets are finite disjoint unions of $C^1$--smooth compact manifolds of dimension $n-1$. These results remain valid also within the larger space of functions of bounded variation $BV_{n}({\mathbb R}^{n})$. For the proofs we establish and use some new results on Luzin--type approximation of Sobolev and BV--functions by $C^k$--functions, where the exceptional sets have small Hausdorff content.
On equality of the $L^\infty$ norm of the gradient under the Hausdorff and Lebesgue measure
本文证明对于定义在 $\mathbb R^n$ 开子集上的函数，其梯度在 $k$ 维 Hausdorff 测度下的 $L^\infty$ 范数等于在 $n$ 维 Lebesgue 测度下的范数，并推出 Sobolev 空间 $W^{1,\infty}$ 中的收敛保持处处可微性。
We prove that if the Hausdorff dimension of a compact subset of ${\mathbb R}^d$ is greater than $\frac{d+1}{2}$, then the set of angles determined by triples of points from this set has positive Lebesgue measure. Sobolev bounds for bi-linear analogs of generalized Radon transforms and the method of stationary phase play a key role. These results complement those of V. Harangi, T. Keleti, G. Kiss, P. Maga, P. Mattila and B. Stenner in (\cite{HKKMMS10}). We also obtain new upper bounds for the number of times an angle can occur among $N$ points in ${\mathbb R}^d$, $d \ge 4$, motivated by the res…
It is known that functions in a Sobolev space with critical exponent embed into the space of functions of bounded mean oscillation, and therefore satisfy the John-Nirenberg inequality and a corresponding exponential integrability estimate. While these inequalities are optimal for general functions of bounded mean oscillation, the main result of this paper is an improvement for functions in a class of critical Sobolev spaces. Precisely, we prove the inequality \[\mathcal{H}^β_{\infty}(\{x\in Ω:|I_αf(x)|>t\})\leq Ce^{-ct^{q'}}\] for all $\|f\|_{L^{N/α,q}(Ω)}\leq 1$ and any $β\in (0,N]$, where $Ω…
We investigate Fourier multipliers with smooth symbols defined over locally compact Hausdorff groups. Our main results in this paper establish new Hörmander-Mikhlin criteria for spectral and non-spectral multipliers. The key novelties which shape our approach are three. First, we control a broad class of Fourier multipliers by certain maximal operators in noncommutative $L_p$ spaces. This general principle ---exploited in Euclidean harmonic analysis during the last 40 years--- is of independent interest and might admit further applications. Second, we replace the formerly used cocycle dimensio…
Rigidity of reflectionless Riesz measures and Sobolev estimates for Riesz transforms on Lipschitz graphs
本文针对Xavier Tolsa提出的正无反射测度刚性猜想，在多项式增长和单点大尺度下质量界条件下，证明了两类平坦性结果，并建立了Lipschitz图上Riesz变换的Sobolev估计。
It is a classical result that Lebesgue measure on the unit circle is invariant under inner functions fixing the origin. In this setting, the distortion of Hausdorff contents has also been studied. We present here similar results focusing on inner functions with fixed points on the unit circle. In particular, our results yield information not only on the size of preimages of sets under inner functions, but also on their distribution with respect to a given boundary point. As an application, we use them to estimate the size of irregular points of inner functions omitting large sets. Finally, we …
On the dimension distortion under fractionally smooth mappings
该研究确定了分数阶光滑连续映射在度量空间间对豪斯多夫、上闵可夫斯基及上中间维数的失真程度，扩展至加权欧几里得空间和加倍度量测度空间，并量化了拟对称映射的维数失真性质。
Continuity for Sobolev mappings with null Lagrangian bounds
本文证明了满足零拉格朗日量型微分不等式界的Sobolev函数的连续性，并作为主要应用，针对带缺陷畸变不等式给出了一个全新的尖锐连续性定理，其证明依赖于以超水平集测度表述的Sobolev型不等式。
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