Related papers: Traces for fractional Sobolev spaces with variable…
We show that, when $sp>N$, the sharp Hardy constant $\mathfrak{h}_{s,p}$ of the punctured space $\mathbb R^N\setminus\{0\}$ in the Sobolev-Slobodecki\u{\i} space provides an optimal lower bound for the Hardy constant…
We extend the well-known result that any $f \in W^{1,n}(\Omega,\mathbb{R}^n)$, $\Omega \subset \mathbb{R}^n$ with strictly positive Jacobian is actually continuous: it is also true for fractional Sobolev spaces $W^{s,\frac{n}{s}}(\Omega)$…
We survey a few trace theorems for Sobolev spaces on $N$-dimensional Euclidean domains. We include known results on linear subspaces, in particular hyperspaces, and smooth boundaries, as well as less known results for Lipschitz boundaries,…
In this article, we study the existence/multiplicity results for the following variable order nonlocal Choquard problem with variable exponents \begin{equation*} \begin{array}{rl}…
Fractional operators are widely used in mathematical models describing abnormal and nonlocal phenomena. Although there are extensive numerical methods for solving the corresponding model problems, theoretical analysis such as the regularity…
In this paper presents the results obtained in the field of spectral theory operators of fractional differentiation. Proven a number of propositions which represents independent interest in the theory of fractional calculus. Introduced…
We introduce a non-linear criterion which allows us to determine when a function can be written as a sum of functions belonging to homogeneous fractional spaces: for $\ell \in \mathbb{N}^*$, $s_i\in (0, 1)$ and $p_i \in [1, +\infty)$, $u :…
In the Euclidean setting the Sobolev spaces $W^{\alpha,p}\cap L^\infty$ are algebras for the pointwise product when $\alpha>0$ and $p\in(1,\infty)$. This property has recently been extended to a variety of geometric settings. We produce a…
A Bourgain--Brezis--Mironescu-type theorem for fractional Sobolev spaces with variable exponents is established for sufficiently regular functions. We prove, however, that a limiting embedding theorem for these spaces fails to hold in…
We present weighted Sobolev spaces and prove a trace theorem for the spaces. As an application, we discuss non-zero boundary value problems for parabolic equations. The weighted parabolic Sobolev spaces we consider are designed, in…
We give a sharp characterization of how additional integrability in the interior improves the integrability of boundary traces of $\mathrm{W}^{1,p}$-Sobolev functions. The optimality of our results relies on a novel nonlinear extension or…
We prove that for $p\ge 2$ solutions of equations modeled by the fractional $p$-Laplacian improve their regularity on the scale of fractional Sobolev spaces. Moreover, under certain precise conditions, they are in $W^{1,p}_{loc}$ and their…
Graph-based analysis holds both theoretical and applied significance, attracting considerable attention from researchers and yielding abundant results in recent years. However, research on fractional problems remains limited, with most of…
In this paper we study approximations of functions of Sobolev spaces $W^2_{p,\loc}(\Omega)$, $\Omega\subset\mathbb R^n$, by Lipschitz continuous functions. We prove that if $f\in W^2_{p,\loc}(\Omega)$, $1\leq p<\infty$, then there exists a…
We establish a family of sharp Sobolev trace inequalities involving the $W^{k,2}(\mathbb{R}_+^{n+1},y^a)$-norm. These inequalities are closely related to the realization of fractional powers of the Laplacian on…
We provide a simplified proof of the existence, under some assumptions, of a spectral gap for the Perron-Frobenius operator of piecewise uniformly expanding maps on Riemannian manifolds when acting on some Sobolev spaces. Its consequences…
Some inequalities for quantum f-divergence of trace class operators in Hilbert spaces are obtained. It is shown that for normalised convex functions it is nonnegative. Some upper bounds for quantum f-divergence in terms of variational and…
In this paper, we study the stability of fractional Sobolev trace inequality within both the functional and critical point settings. In the functional setting, we establish the following sharp estimate:…
We continue the study of the space $BV^\alpha(\mathbb R^n)$ of functions with bounded fractional variation in $\mathbb R^n$ and of the distributional fractional Sobolev space $S^{\alpha,p}(\mathbb R^n)$, with $p\in [1,+\infty]$ and…
We study an inverse problem for variable coefficient fractional parabolic operators of the form $(\partial_t -\operatorname{div}(A(x) \nabla_x)^s + q(x,t)$ for $s\in(0,1)$ and show the unique recovery of $q$ from exterior measured data.…