Related papers: (1,p)-Sobolev spaces based on strongly local Diric…

We prove Rellich-Kondrachov type theorems on the half-space $\mathbb{H}^{N+1}=\{(y, x) \in \left.\mathbb{R} \times \mathbb{R}^N: y>0\right\}$ endowed with the general weighted measure $\mu_w:=y^c \phi(|z|) d z$, where $c \in \mathbb{R}$ and…

We investigate weighted Sobolev spaces on metric measure spaces $(X,d,m)$. Denoting by $\rho$ the weight function, we compare the space $W^{1,p}(X,d,\rho m)$ (which always concides with the closure $H^{1,p}(X,d,\rho m)$ of Lipschitz…

We compute the precise value of the measure of noncompactness of Sobolev embeddings $W_0^{1,p}(D)\hookrightarrow L^p(D)$, $p\in(1,\infty)$, on strip-like domains $D$ of the form $\mathbb{R}^k\times\prod\limits_{i=1}^{n-k}(a_i,b_i)$. We show…

We give the definition of $L^p$-convergence of tensor fields with respect to the Gromov-Hausdorff topology and several fundamental properties of the convergence. We apply this to establish a Bochner-type inequality which keeps the term of…

We introduce a generalization of relative entropy derived from the Wigner-Yanase-Dyson entropy and give a simple, self-contained proof that it is convex. Moreover, special cases yield the joint convexity of relative entropy, and for the map…

We consider 1-complemented subspaces (ranges of contractive projections) of vector-valued spaces $\ell_p(X)$, where $X$ is a Banach space with a 1-unconditional basis and $p \in (1,2)\cup (2,\infty)$. If the norm of $X$ is twice…

We study removable sets for Newtonian Sobolev functions in metric measure spaces satisfying the usual (local) assumptions of a doubling measure and a Poincar\'e inequality. In particular, when restricted to Euclidean spaces, a closed set…

The seminal paper of Daubechies, Defrise, DeMol made clear that $\ell^p$ spaces with $p\in [1,2)$ and $p$-powers of the corresponding norms are appropriate settings for dealing with reconstruction of sparse solutions of ill-posed problems…

In this paper, we extend the fractional Sobolev spaces with variable exponents $W^{s,p(x,y)}$ to include the general fractional case $W^{K,p(x,y)}$, where $p$ is a variable exponent, $s\in (0,1)$ and $K$ is a suitable kernel. We are…

We introduce a new functional $\mathcal{E}_{\mathfrak{p}}$ on the space of conformal structures on an oriented projective manifold $(M,\mathfrak{p})$. The nonnegative quantity $\mathcal{E}_{\mathfrak{p}}([g])$ measures how much…

To describe a set of functions, which forms a reflexive subspace B of the classical Banach space L a special function that characterizes their average integral growth is introduced. It is shown that this function essentially depends on the…

Given $1<p<N$ and two measurable functions $V\left( r\right) \geq 0$ and $K\left( r\right) >0$, $r>0$, we define the weighted spaces \[ W=\left\{ u\in D^{1,p}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}V\left( \left| x\right| \right) \left|…

We identify a large class of constant (complex) coefficient, second order elliptic systems for which the Dirichlet problem in the upper-half space with data in $L^p$-based Sobolev spaces, $1<p<\infty$, of arbitrary smoothness $\ell$, is…

We derive a discrete $ L^q-L^p$ Sobolev inequality tailored for the Crouzeix--Raviart and discontinuous Crouzeix--Raviart finite element spaces on anisotropic meshes in both two and three dimensions. Subject to a semi-regular mesh…

We show that a dynamically convex Reeb flow on the standard tight lens space $(L(p, 1),\xi_{\mathrm{std}})$, $p>1,$ admits a $p$-unknotted closed Reeb orbit $P$ which is the binding of a rational open book decomposition with disk-like…

The aim of this work is to study the first order Dirac-Sobolev spaces in $L^p$ norm on an open subset of ${\mathbb R}^3$ to clarify its relationship with the corresponding Sobolev spaces. It is shown that for $1< p <\infty$, they coincide,…

We consider finite energy and $L^2$ differential forms associated with strongly local regular Dirichlet forms on compact connected topologically one-dimensional spaces. We introduce notions of local exactness and local harmonicity and prove…

We investigate Sobolev spaces $W^{1,\Phi}$ associated to Musielak-Orlicz spaces $L^\Phi$. We first present conditions for the boundedness of the Voltera operator in $L^\Phi$. Employing this, we provide necessary and sufficient conditions…

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…

On a compact connected group $G$, consider the infinitesimal generator $-L$ of a central symmetric Gaussian convolution semigroup $(\mu_t)_{t>0}$. We establish several regularity results of the solution to the Poisson equation $LU=F$, both…