Related papers: Quality of non-compactness for Sobolev Embedding w…

Interpolation inequalities play an important role in the study of PDEs and their applications. There are still some interesting open questions and problems that related to integral estimates and regularity of solutions to the elliptic…

Given $0<p,q, r<\infty $ and $ q<r\le \infty$ we consider the natural embedding $\ell_{p,q}\hookrightarrow \ell_{p,r}$ between Lorenz sequence spaces. We prove that this non-compact embedding is always strictly singular but not finitely…

In this article, we study local Sobolev-Poincar\'e imbedding domains. The main result reads as below. \begin{enumerate} \item for $1\leq p\leq n$, a bounded uniform domain is also a local Sobolev-Poincar\'e imbedding domain of order $p$;…

We investigate the size of fixed point sets of automorphisms of bounded domains in $\mathbb{C}^n$. In one complex variable, a nontrivial automorphism has at most two fixed points, but in higher dimensions fixed point sets need not be…

In this article, we generalize Eberlein's Rigidity Theorem to the singular case, namely, one of the spaces is only assumed to be a CAT(0) topological manifold. As a corollary, we get that any compact irreducible but locally reducible…

Non-locality is being intensively studied in various PDE-contexts and in variational problems. The numerical approximation also looks challenging, as well as the application of these models to Continuum Mechanics and Image Analysis, among…

We study integration and $L_2$-approximation on countable tensor products of function spaces of increasing smoothness. We obtain upper and lower bounds for the minimal errors, which are sharp in many cases including, e.g., Korobov, Walsh,…

This paper presents two general criteria to determine spaceability results in the complements of unions of subspaces. The first criterion applies to countable unions of subspaces under specific conditions and is closely related to the…

We prove local and global invertibility of Sobolev solutions of certain differential inclusions which prevent the differential matrix from having negative eigenvalues. Our results are new even for quasiregular mappings in two dimensions.

We deal with Orlicz-Sobolev embeddings in open subsets of $\mathbb{R}^n$. A necessary and sufficient condition is established for the existence of an optimal, i.e. largest possible, Orlicz-Sobolev space continuously embedded into a given…

The tensorization problem for Sobolev spaces asks for a characterization of how the Sobolev space on a product metric measure space $X\times Y$ can be determined from its factors. We show that two natural descriptions of the Sobolev space…

We consider extremal polynomials with respect to a Sobolev-type $p$-norm, with $1<p<\infty$ and measures supported on compact subsets of the real line. For a wide class of such extremal polynomials with respect to mutually singular measures…

For a class of density functions $q^n(x^n)$ on $\Bbb R^n$ we prove an inequality between relative entropy and the sum of average conditional relative entropies of the following form: For any density function $p^n(x^n)$ on $\Bbb R^n$,…

In the present paper, we investigate whether an embedding of a decomposition space $\mathcal{D}\left(\mathcal{Q},L^{p},Y\right)$ into a given Sobolev space $W^{k,q}(\mathbb{R}^{d})$ exists. As special cases, this includes embeddings into…

We study the removability of a singular set in the boundary of Neumann problem for elliptic equations with variable exponent. We consider the case where the singular set is compact, and give sufficient conditions for removability of this…

We recall some known and present several new results about Sobolev spaces defined with respect to a measure, in particular a precise pointwise description of the tangent space to this measure in dimension 1. This allows to obtain an…

We construct a family of non-parametric (infinite-dimensional) manifolds of finite measures on $R^d$. The manifolds are modelled on a variety of weighted Sobolev spaces, including Hilbert-Sobolev spaces and mixed-norm spaces. Each supports…

We establish equivalence between the boundedness of specific supremum operators and the optimality of function spaces in Sobolev embeddings acting on domains in ambient Euclidean space with a prescribed isoperimetric behavior. Our approach…

Let $\Omega\subset\mathbb{R}^n$ be an open set and let $f\in W^{1,p}(\Omega,\mathbb{R}^n)$ be a weak (sequential) limit of Sobolev homeomorphisms. Then $f$ is injective almost everywhere for $p>n-1$ both in the image and in the domain. For…

We obtain a compact Sobolev embedding for $H$-invariant functions in compact metric-measure spaces, where $H$ is a subgroup of the measure preserving bijections. In Riemannian manifolds, $H$ is a subgroup of the volume preserving…