Related papers: A new approach to Sobolev spaces in metric measure…

In this paper, we develop the theory of Sobolev spaces on locally finite graphs, including completeness, reflexivity, separability, and Sobolev inequalities. Since there is no exact concept of dimension on graphs, classical methods that…

Let $X$ be a quasi-Banach space of analytic functions in the unit disc and let $q>0$. A finite positive Borel measure $\mu$ in the closed unit disc $\overline{\mathbb{D}}$ is called a $q$-reverse Carleson measure for $X$ if and only if…

For a positive finite Borel measure $\mu$ compactly supported in the complex plane, the space $\mathcal{P}^2(\mu)$ is the closure of the analytic polynomials in the Lebesgue space $L^2(\mu)$. According to Thomson's famous result, any space…

Optimal higher-order Sobolev type embeddings are shown to follow via isoperimetric inequalities. This establishes a higher-order analogue of a well-known link between first-order Sobolev embeddings and isoperimetric inequalities. Sobolev…

We introduce and study fractional variable exponents Sobolev trace spaces on any open set in the Euclidean space equipped with the Lebesgue measure. We show that every equivalence class of Sobolev functions has a quasicontinuous…

Let $(X, d)$ be a compact metric space and let $\mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I \colon \mathcal{M}(X) \to \R$ by \[I(\mu) = \int_X \int_X d(x,y) d\mu(x) d\mu(y),\] and set $M(X) = \sup…

We prove that if $X$ is a real rearrangement-invariant function space on $[0,1]$, which is not isometrically isomorphic to $L_2,$ then every surjective isometry $T:X\to X$ is of the form $Tf(s)=a(s)f(\sigma(s))$ for a Borel function $a$ and…

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…

Let $\mu$ be a positive finite measure on the unit circle. The Dirichlet type space $\mathcal{D}(\mu)$, associated to $\mu$, consists of holomorphic functions on the unit disc whose derivatives are square integrable when weighted against…

In this article, we introduce classes of functions whose increment is controlled by the measure of a ball containing the corresponding points and a nonnegative function p(.) that is summable with respect to measure. These classes of…

In this paper we show that if $(X,\mathcal{A})$ is a measurable space and if $Y$ is a topological model of a Lawvere theory $\mathcal{T}$ equipped with $\mathcal{B}$ the Borel $\sigma$-algebra on $Y$, then the set of…

This paper studies smoothing properties of the local fractional maximal operator, which is defined in a proper subdomain of the Euclidean space. We prove new pointwise estimates for the weak gradient of the maximal function, which imply…

We propose a functional framework of fractional Sobolev spaces for a class of ultra-parabolic Kolmogorov type operators satisfying the weak H\"ormander condition. We characterize these spaces as real interpolation of natural order intrinic…

Let $\mu$ be a Borel measure on a compactum $X$. The main objects in this paper are $\sigma$-ideals $I(dim)$, $J_0(\mu)$, $J_f(\mu)$ of Borel sets in $X$ that can be covered by countably many compacta which are finite-dimensional, or of…

Let $\gamma\in\mathbb{R}\setminus\{0\}$ and $X(\mathbb{R}^n)$ be a ball Banach function space satisfying some extra mild assumptions. Assume that $\Omega=\mathbb{R}^n$ or $\Omega\subset\mathbb{R}^n$ is an $(\varepsilon,\infty)$-domain for…

It is well known that functions in the analytic Besov space $B_1$ on the unit disk $\D$ admits an integral representation $$f(z)=\ind\frac{z-w}{1-z\bar w}\,d\mu(w),$$ where $\mu$ is a complex Borel measure with $|\mu|(\D)<\infty$. We…

The purpose of this investigation is to extend basic equations and inequalities which hold for functions $f$ in a Bernstein space $B_\sigma^2$ to larger spaces by adding a remainder term which involves the distance of $f$ from $B_\sigma^2$.…

For the full range of index $0<p\leq\infty$, real weight $\alpha$ and real Sobolev order $s$, two types of weighted Fock-Sobolev spaces over $\mathbb C^n$, $F^p_{\alpha, s}$ and $\widetilde F^p_{\alpha,s}$, are introduced through fractional…

Let $\mu$ be a Borel probability measure on a compact path-connected metric space $(X, \rho)$ for which there exist constants $c,\beta>1$ such that $\mu(B) \geq c r^{\beta}$ for every open ball $B\subset X$ of radius $r>0$. For a class of…

We prove general results about separation and weak$^\#$-convergence of boundedly finite measures on separable metric spaces and Souslin spaces. More precisely, we consider an algebra of bounded real-valued, or more generally a $*$-algebra…