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Search: MSC category 46E35 ( Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems )

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1. CJM Online first

Wang, Xing; Zhang, Chunjie
Pointwise convergence of solutions to the Schrödinger equation on manifolds
Let $(M^n,g)$ be a Riemannian manifold without boundary. We study the amount of initial regularity is required so that the solution to free Schrödinger equation converges pointwise to its initial data. Assume the initial data is in $H^\alpha(M)$. For Hyperbolic Space, standard Sphere and the 2 dimensional Torus, we prove that $\alpha\gt \frac{1}{2}$ is enough. For general compact manifolds, due to lacking of local smoothing effect, it is hard to beat the bound $\alpha\gt 1$ from interpolation. We managed to go below 1 for dimension $\leq 3$. The more interesting thing is that, for 1 dimensional compact manifold, $\alpha\gt \frac{1}{3}$ is sufficient.

Keywords:pointwise convergence, Schrödinger operator, manifold, Strichartz estimate
Categories:35L05, 46E35, 42B37

2. CJM 2013 (vol 66 pp. 721)

Durand-Cartagena, E.; Ihnatsyeva, L.; Korte, R.; Szumańska, M.
On Whitney-type Characterization of Approximate Differentiability on Metric Measure Spaces
We study approximately differentiable functions on metric measure spaces admitting a Cheeger differentiable structure. The main result is a Whitney-type characterization of approximately differentiable functions in this setting. As an application, we prove a Stepanov-type theorem and consider approximate differentiability of Sobolev, $BV$ and maximal functions.

Keywords:approximate differentiability, metric space, strong measurable differentiable structure, Whitney theorem
Categories:26B05, 28A15, 28A75, 46E35

3. CJM 2013 (vol 66 pp. 641)

Grigor'yan, Alexander; Hu, Jiaxin
Heat Kernels and Green Functions on Metric Measure Spaces
We prove that, in a setting of local Dirichlet forms on metric measure spaces, a two-sided sub-Gaussian estimate of the heat kernel is equivalent to the conjunction of the volume doubling propety, the elliptic Harnack inequality and a certain estimate of the capacity between concentric balls. The main technical tool is the equivalence between the capacity estimate and the estimate of a mean exit time in a ball, that uses two-sided estimates of a Green function in a ball.

Keywords:Dirichlet form, heat kernel, Green function, capacity
Categories:35K08, 28A80, 31B05, 35J08, 46E35, 47D07

4. CJM 2007 (vol 59 pp. 1135)

Björn, Anders; Björn, Jana; Shanmugalingam, Nageswari
Sobolev Extensions of Hölder Continuous and Characteristic Functions on Metric Spaces
We study when characteristic and H\"older continuous functions are traces of Sobolev functions on doubling metric measure spaces. We provide analytic and geometric conditions sufficient for extending characteristic and H\"older continuous functions into globally defined Sobolev functions.

Keywords:characteristic function, Newtonian function, metric space, resolutivity, Hölder continuous, Perron solution, $p$-harmonic, Sobolev extension, Whitney covering
Categories:46E35, 31C45

5. CJM 2006 (vol 58 pp. 492)

Chua, Seng-Kee
Extension Theorems on Weighted Sobolev Spaces and Some Applications
We extend the extension theorems to weighted Sobolev spaces $L^p_{w,k}(\mathcal D)$ on $(\varepsilon,\delta)$ domains with doubling weight $w$ that satisfies a Poincar\'e inequality and such that $w^{-1/p}$ is locally $L^{p'}$. We also make use of the main theorem to improve weighted Sobolev interpolation inequalities.

Keywords:Poincaré inequalities, $A_p$ weights, doubling weights, $(\ep,\delta)$ domain, $(\ep,\infty)$ domain

6. CJM 2002 (vol 54 pp. 1280)

Skrzypczak, Leszek
Besov Spaces and Hausdorff Dimension For Some Carnot-Carathéodory Metric Spaces
We regard a system of left invariant vector fields $\mathcal{X}=\{X_1,\dots,X_k\}$ satisfying the H\"ormander condition and the related Carnot-Carath\'eodory metric on a unimodular Lie group $G$. We define Besov spaces corresponding to the sub-Laplacian $\Delta=\sum X_i^2$ both with positive and negative smoothness. The atomic decomposition of the spaces is given. In consequence we get the distributional characterization of the Hausdorff dimension of Borel subsets with the Haar measure zero.

Keywords:Besov spaces, sub-elliptic operators, Carnot-Carathéodory metric, Hausdorff dimension
Categories:46E35, 43A15, 28A78

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