相关论文: Entropy and approximation numbers of weighted Sobo…

This paper studies the Sobolev-Lorentz capacity and its regularity in the Euclidean setting for $n \ge 1$ integer. We extend here our previous results on the Sobolev-Lorentz capacity obtained for $n \ge 2.$ Moreover, for $n \ge 2$ integer…

We consider the order of convergence for linear and nonlinear Monte Carlo approximation of compact embeddings from Sobolev spaces of dominating mixed smoothness defined on the torus $\mathbb{T}^d$ into the space $L_{\infty}(\mathbb{T}^d)$…

Let $R>1$ and let $B$ be the Euclidean $4$-ball of radius $R$ with a closed subset ${E}$ removed. Suppose that $B$ embeds symplectically into the unit cylinder $\mathbb{D}^2 \times \mathbb{R}^2$. By Gromov's non-squeezing theorem, ${E}$…

The paper is dedicated to the study of embeddings of the anisotropic Besov spaces $B^{\beta_1,...,beta_n}_{p;\theta_1,...,\theta_n}(\Bbb R^n)$ into Lorentz spaces. We find the sharp asymptotic behaviour of embedding constants when some of…

In this paper, we investigate optimal linear approximations ($n$-approximation numbers ) of the embeddings from the Sobolev spaces $H^r\ (r>0)$ for various equivalent norms and the Gevrey type spaces $G^{\alpha,\beta}\ (\alpha,\beta>0)$ on…

The article deals with a simplified proof of the Sobolev embedding theorem for Lizorkin--Triebel spaces (that contain the $L_p$-Sobolev spaces $H^s_p$ as special cases). The method extends to a proof of the corresponding fact for general…

For a separable rearrangement invariant space $X$ on $(0,\infty)$ of fundamental type we identify the set of all $p\in [1,\infty]$ such that $\ell^p$ is finitely represented in $X$ in such a way that the unit basis vectors of $\ell^p$…

Motivated by the Bekenstein Hawking formula and the area law behaviour of entanglement entropy, we propose that in any UV finite theory of quantum gravity with a smooth spacetime, the total entropy for a pure state in a co-dimension one…

In this paper we obtain logarithmic corrections to the black hole entropy. Motivated by our recent proposal concerning the nature of the degrees of freedom leading to the black hole entropy in terms of a Bose Einstein (BEC) condensate of…

We prove uniform boundedness of certain boundary representations on appropriate fractional Sobolev spaces $W^{s,p}$ with $p>1$ for arbitrary Gromov hyperbolic groups. These are closed subspaces of $L^p$ and in particular Hilbert spaces in…

Let $L_{q,\mu}$, $1\leq q\leq\infty$, denotes the weighted $L_q$ space of functions on the unit ball $\Bbb B^d$ with respect to weight $(1-\|x\|_2^2)^{\mu-\frac12},\,\mu\ge 0$, and let $W_{2,\mu}^r$ be the weighted Sobolev space on $\Bbb…

The variational capacity cap_p in Euclidean spaces is known to enjoy the density dichotomy at large scales, namely that for every subset E of R^n, inf_{x in R^n} (cap_p(E \cap B(x,r),B(x,2r)) / cap_p(B(x,r),B(x,2r))) is either zero or tends…

We investigate properties of the pseudo-Riemannian volume, entropy, and diameter for convex cocompact representations $\rho : \Gamma \to \mathrm{SO}(p,q+1)$ of closed $p$-manifold groups. In particular: We provide a uniform lower bound of…

Extremal black holes in M-theory compactification on $M\times S^1$ are microscopically represented by fivebranes wrapping $P\times S^1$, where $M$ is a Calabi-Yau threefold and $P$ is a four-cycle in $M$. Additional spacetime charges arise…

It is known that if finite subsets of a locally finite metric space $M$ admit $C$-bilipschitz embeddings into $\ell_p$ $(1\le p\le \infty)$, then for every $\epsilon>0$, the space $M$ admits a $(C+\epsilon)$-bilipschitz embedding into…

The Riemann Mapping Theorem states existence of a conformal homeomorphism $\varphi$ of a simply connected plane domain $\Omega\subset\mathbb C$ with non-empty boundary onto the unit disc $\mathbb D\subset \mathbb C$. In the first part of…

Similarity-sensitive entropy measures the uncertainty of a probability law relative to a similarity kernel that encodes the distinguishability between states. We develop a measure-theoretic treatment covering both finite similarity matrices…

Using martingale methods, we provide bounds for the entropy of a probability measure on $\mathbb {R}^d$ with the right-hand side given in a certain integral form. As a corollary, in the one-dimensional case, we obtain a weighted log-Sobolev…

We investigate the existence of embeddings of shearlet coorbit spaces associated to weighted mixed $L^p$-spaces into classical Sobolev spaces in dimension three by using the description of coorbit spaces as decomposition spaces. This…

The purpose of this short article is to prove some potential estimates that naturally arise in the study of subelliptic Sobolev inequalites for functions. This will allow us to prove a local subelliptic Sobolev inequality with the optimal…