Related papers: Fractional Sobolev spaces with kernel function on …

We consider a broad class of nonlinear integro-differential equations with a kernel whose differentiability order is described by a general function $\phi$. This class includes not only the fractional $p$-Laplace equations, but also…

It is common that a Sobolev space defined on $\mathbb{R}^m$ has a non-compact embedding into an $L^p$-space, but it has subspaces for which this embedding becomes compact. There are three well known cases of such subspaces, the Rellich…

We establish necessary and sufficient conditions guaranteeing compactness of embeddings of fractional Sobolev spaces, Besov spaces, and Triebel-Lizorkin spaces, in the general context of quasi-metric-measure spaces. Although stated in the…

This article is devoted to developing a theory for effective kernel interpolation and approximation in a general setting. For a wide class of compact, connected $C^\infty$ Riemannian manifolds, including the important cases of spheres and…

This paper is concerned with the study of a nonlinear problems involving the fractional p(x)-Laplacian operator. By means of the Berkovits degree theory, we prove the existence of nontrivial weak solutions for this problem. The appropriate…

We prove that the geodesic equations of all Sobolev metrics of fractional order one and higher on spaces of diffeomorphisms and, more generally, immersions are locally well posed. This result builds on the recently established real analytic…

We construct fractional Sobolev spaces on arbitrary time scales, both in one dimension and on product time scales. In 1D, we define $W^{\alpha(\cdot),p}_{\mathrm{rd}}(\mathcal I)$ through a variable-order Gagliardo-type seminorm and prove…

We give an elementary proof of a compact embedding theorem in abstract Sobolev spaces. The result is first presented in a general context and later specialized to the case of degenerate Sobolev spaces defined with respect to nonnegative…

We provide a complete characterization of compactness of Sobolev embeddings of radially symmetric functions on the entire space $\mathbb{R}^n$ in the general framework of rearrangement-invariant function spaces. We avoid any unnecessary…

We define and study homogeneous kinetic Sobolev spaces adapted to the Kolmogorov equation. We consider both local and non-local diffusion. The spaces are built from the Lebesgue spaces L p for all integrability exponents p $\in$ (1,…

This short note investigates the compact embedding of degenerate matrix weighted Sobolev spaces into weighted Lebesgue spaces. The Sobolev spaces explored are defined as the abstract completion of Lipschitz functions in a bounded domain…

When analyzing modern machine learning algorithms, we may need to handle kernel density estimation (KDE) with intricate kernels that are not designed by the user and might even be irregular and asymmetric. To handle this emerging challenge,…

The aim of this work is to prove a compact embedding for a weighted fractional Sobolev spaces. As an application, we use this embedding to prove, via variational methods, the existence of solutions for the following Schr\"odinger equation…

In this work, we establish some abstract results on the perspective of the fractional Musielak-Sobolev spaces, such as: uniform convexity, Radon-Riesz property with respect to the modular function, $(S_{+})$-property, Brezis-Lieb type Lemma…

The purpose of this paper is to extend the embedding theorem of Sobolev spaces involving general kernels and we provide a sharp critical exponent in these embeddings. As an application, solutions for equations driven by a general…

The objective of this paper is twofold. First, we conduct a careful study of various functional inequalities involving the fractional Laplacian operators, including nonlocal Sobolev-Poincar\'e, Nash, Super Poincar\'e and logarithmic Sobolev…

Given a compact metric graph $\Gamma$ and the Laplacian $\Delta_{\Gamma}$ coupled with standard (Kirchhoff) vertex conditions, solutions to fractional elliptic partial differential equations of the form $(\kappa^2 -…

Given a complete non-compact Riemannian manifold $(M,g)$ with certain curvature restrictions, we introduce an expansion condition concerning a group of isometries $G$ of $(M,g)$ that characterizes the coerciveness of $G$ in the sense of…

This paper is a self-contained presentation of certain aspects of the theory of weighted Sobolev spaces and elliptic operators on non-compact Riemannian manifolds. Specifically, we discuss (i) the standard and weighted Sobolev Embedding…

It is shown that most of the well-known basic results for Sobolev-Slobodeckii and Bessel potential spaces, known to hold on bounded smooth domains in $\mathbb{R}^n$, continue to be valid on a wide class of Riemannian manifolds with…