Related papers: Trace and extension theorems for Sobolev-type func…

This article is a continuation of our work on generalized matrix-weighted Besov--Triebel--Lizorkin-type spaces with matrix $\mathcal{A}_{\infty}$ weights. In this article, we establish the boundedness of pseudo-differential, trace, and…

We first introduce an appropriate family of conformally covariant boundary operators associated to the Siegel domain ${\mathcal U}^{n+1}$ with the Heisenberg group $\mathbb{H}^{n}$ as its boundary and the complex ball…

We study limiting trace inequalities in the style of Maz'ya and Meyers--Ziemer for Sobolev martingales. We develop the Bellman function approach to such estimates, which allows to provide sufficient and almost necessary conditions on the…

Let $(\operatorname{X},\operatorname{d},\mu)$ be a metric measure space with uniformly locally doubling measure $\mu$. Given $p \in (1,\infty)$, assume that $(\operatorname{X},\operatorname{d},\mu)$ supports a weak local $(1,p)$-Poincar\'e…

We give characterizations for the existence of traces for first order Sobolev spaces defined on regular trees.

This paper is concerned with a boundedness of trace and extension operators for Besov spaces and Triebel-Lizorkin spaces on upper half space with variable exponents. To define trace and extension operators, we introduce a quarkonial…

It is a classical theorem that if a function is integrable along the boundary of the unit circle, then the function is the nontangential limit of a holomorphic function on the open disc if and only if its Fourier coefficients for…

Sobolev-type embeddings on metric measure spaces encode a subtle interaction between the analytic regularity of functions and the geometry of the underlying domain space. In this paper we develop an embedding theory for variable…

We study a characterization of BV and Sobolev functions via nonlocal functionals in metric spaces equipped with a doubling measure and supporting a Poincar\'e inequality. Compared with previous works, we consider more general functionals.…

A new notion of a Hausdorff-type operator on function spaces over domains in Euclidean spaces is introduced, and a sufficient condition for the boundedness of this operator on Sobolev spaces is proved. It is shown that this condition cannot…

This paper deals with the fractional Sobolev spaces W^[s,p]. We analyze the relations among some of their possible definitions and their role in the trace theory. We prove continuous and compact embeddings, investigating the problem of the…

In this paper we obtain some practical criteria to bound the multiplication operator in Sobolev spaces with respect to measures in curves. As a consequence of these results, we characterize the weighted Sobolev spaces with bounded…

In this paper, we solve the Dirichlet problem for Sobolev maps between singular metric spaces that extends the corresponding result of Guo and Wenger [Comm. Anal. Geom. 2020]. The main new ingredient in our proofs is a suitable extension of…

We study comprehensively local properties of functions in complex Sobolev spaces on a bounded open subset of $\mathbb{C}^n$. The main tool is the corresponding functional capacity for the space which is inspired by the global one due to…

We prove that a trace inequality holds for John domains $\Omega$ satisfying $$ \mathcal H^{n-1}(\partial \Omega\setminus \partial_*\Omega)=0,$$ where $\partial_*\Omega$ denotes the measure-theoretic boundary, together with an upper density…

In a doubling metric measure space $(X,\rho,\mu)$ supporting a Poincar\'e inequality, we give a new characterisation of first-order Sobolev spaces by mean oscillations, and extend previous characterisations of constant functions in terms of…

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…

We consider integrals in the sense of Choquet with respect to the $\delta$-dimensional Hausdorff content for continuously differentiable functions defined on open, connected sets in the Euclidean $n$-space, $n\geq 2$, $0<\delta\le n$. In…

This is the second of two works concerning the Sobolev calculus on metric measure spaces and its applications. In this work, we focus on several approaches to vector calculus in the non-smooth setting of complete and separable metric spaces…

In this paper we make a survey of some recent developments of the theory of Sobolev spaces $W^{1,q}(X,\sfd,\mm)$, $1<q<\infty$, in metric measure spaces $(X,\sfd,\mm)$. In the final part of the paper we provide a new proof of the…