Traces of some weighted function spaces and related non‐standard real interpolation of Besov spaces

We study traces of weighted Triebel–Lizorkin spaces F p , q s ( R n , w ) $F^s_{p,q}(\mathbb {R}^n,w)$ on hyperplanes R n − k $\mathbb {R}^{n-k}$ , where the weight is of Muckenhoupt type. We concentrate on the example weight w α ( x ) = | x n | α $w_\alpha (x) = {\big\vert x_n\big\vert }^\alpha$ when | x n | ≤ 1 $\big\vert x_n\big\vert \le 1$ , x ∈ R n $x\in \mathbb {R}^n$ , and w α ( x ) = 1 $w_\alpha (x)=1$ otherwise, where α > − 1 $\alpha >-1$ . Here we use some refined atomic decomposition argument as well as an appropriate wavelet representation in corresponding (unweighted) Besov spaces. The second main outcome is the description of the real interpolation space ( B p 1 , p 1 s 1 ( R n − k ) , B p 2 , p 2 s 2 ( R n − k ) ) θ , r $\big (B^{s_1}_{p_1,p_1}\big (\mathbb {R}^{n-k}\big ), B^{s_2}_{p_2,p_2}{\big (\mathbb {R}^{n-k}\big )\big )}_{\theta ,r}$ , 0 0$sufficiently large, 0 < θ < 1$0<\theta <1$, 0 < r ≤ ∞$

Interpolation of Hilbert and Sobolev Spaces: Quantitative Estimates and Counterexamples

This paper provides an overview of interpolation of Banach and Hilbert spaces, with a focus on establishing when equivalence of norms is in fact equality of norms in the key results of the theory. (In brief, our conclusion for the Hilbert space case is that, with the right normalisations, all the key results hold with equality of norms.) In the final section we apply the Hilbert space results to the Sobolev spaces $H^s(\Omega)$ and $\widetilde{H}^s(\Omega)$, for $s\in \mathbb{R}$ and an open $\Omega\subset \mathbb{R}^n$. We exhibit examples in one and two dimensions of sets $\Omega$ for which these scales of Sobolev spaces are not interpolation scales. In the cases when they are interpolation scales (in particular, if $\Omega$ is Lipschitz) we exhibit examples that show that, in general, the interpolation norm does not coincide with the intrinsic Sobolev norm and, in fact, the ratio of these two norms can be arbitrarily large

A Probabilistic proof of the breakdown of Besov regularity in LL-shaped domains

{We provide a probabilistic approach in order to investigate the smoothness of the solution to the Poisson and Dirichlet problems in $L$-shaped domains. In particular, we obtain (probabilistic) integral representations for the solution. We also recover Grisvard's classic result on the angle-dependent breakdown of the regularity of the solution measured in a Besov scale

Medial Features for Superpixel Segmentation

Image segmentation plays an important role in computer vision and human scene perception. Image oversegmentation is a common technique to overcome the problem of managing the high number of pixels and the reasoning among them. Specifically, a local and coherent cluster that contains a statistically homogeneous region is denoted as a superpixel. In this paper we propose a novel algorithm that segments an image into superpixels employing a new kind of shape centered feature which serve as a seed points for image segmentation, based on Gradient Vector Flow fields (GVF) [14]. The features are located at image locations with salient symmetry. We compare our algorithm to state-of-the-art superpixel algorithms and demonstrate a performance increase on the standard Berkeley Segmentation Dataset

On Bogovski\u{\i} and regularized Poincar\'e integral operators for de Rham complexes on Lipschitz domains

We study integral operators related to a regularized version of the classical Poincar\'e path integral and the adjoint class generalizing Bogovski\u{\i}'s integral operator, acting on differential forms in $R^n$. We prove that these operators are pseudodifferential operators of order -1. The Poincar\'e-type operators map polynomials to polynomials and can have applications in finite element analysis. For a domain starlike with respect to a ball, the special support properties of the operators imply regularity for the de Rham complex without boundary conditions (using Poincar\'e-type operators) and with full Dirichlet boundary conditions (using Bogovski\u{\i}-type operators). For bounded Lipschitz domains, the same regularity results hold, and in addition we show that the cohomology spaces can always be represented by $C^\infty$ functions.Comment: 23 page

Stability of complex hyperbolic space under curvature-normalized Ricci flow

Using the maximal regularity theory for quasilinear parabolic systems, we prove two stability results of complex hyperbolic space under the curvature-normalized Ricci flow in complex dimensions two and higher. The first result is on a closed manifold. The second result is on a complete noncompact manifold. To prove both results, we fully analyze the structure of the Lichnerowicz Laplacian on complex hyperbolic space. To prove the second result, we also define suitably weighted little H\"{o}lder spaces on a complete noncompact manifold and establish their interpolation properties.Comment: Some typos in version 2 are correcte

Local regularity for fractional heat equations

We prove the maximal local regularity of weak solutions to the parabolic problem associated with the fractional Laplacian with homogeneous Dirichlet boundary conditions on an arbitrary bounded open set $\Omega\subset\mathbb{R}^N$. Proofs combine classical abstract regularity results for parabolic equations with some new local regularity results for the associated elliptic problems.Comment: arXiv admin note: substantial text overlap with arXiv:1704.0756