Related papers: Limits at infinity for Haj{\l}asz-Sobolev function…

We establish the existence and uniqueness of limits at infinity along infinite curves outside a zero modulus family for functions in a homogeneous Sobolev space under the assumption that the underlying space is equipped with a doubling…

In this paper, we study the behaviour at infinity of $p$-Sobolev functions in the setting of Ahlfors $Q$-regular metric measure spaces supporting a $p$-Poincar\'e inequality. By introducing the notions of sets which are $p$-thin at…

We establish optimal results on limits at infinity for functions in fractional Sobolev spaces.

We characterize the restrictions of first order Sobolev functions to regular subsets of a homogeneous metric space and prove the existence of the corresponding linear extension operator.

We study a capacity theory based on a definition of Haj{\l} asz-Besov functions. We prove several properties of this capacity in the general setting of a metric space equipped with a doubling measure. The main results of the paper are lower…

We study necessary and sufficient conditions for a Muckenhoupt weight $w \in L^1_{\mathrm{loc}}(\mathbb R^d)$ that yield almost sure existence of radial, and vertical, limits at infinity for Sobolev functions $u \in…

We study the Sobolev inequality and the existence of its extremal functions in the setting of homogeneous H\"{o}rmander vector fields. A principal result establishes a mutual inclusion between the set of volume growth rates of subunit balls…

In this paper, we study the existence of limits at infinity along almost every infinite curve for the upper and lower approximate limits of bounded variation functions on complete unbounded metric measure spaces. We prove that if the…

The support of wavelet transform associated with square integrable irreducible representation of a homogeneous space is shown to have infinite measure. Pointwise homogeneous approximation property for wavelet transform has been…

We investigate geometric properties of a metric measure space where every function in the Newton--Sobolev space $N^{1,\infty}(Z)$ has a Lipschitz representative. We prove that when the metric space is locally complete and the reference…

In complete metric measure spaces equipped with a doubling measure and supporting a weak Poincar\'e inequality, we investigate when a given Banach-valued Sobolev function defined on a subset satisfying a measure-density condition is the…

We introduce and study two new relations between function spaces over measure spaces of infinite measure, motivated by the question of establishing compactness. The first relation captures the uniform decay of function (quasi-)norms ``at…

Newtonian spaces generalize first-order Sobolev spaces to abstract metric measure spaces. In this paper, we study regularity of Newtonian functions based on quasi-Banach function lattices. Their (weak) quasi-continuity is established,…

We obtain a pointwise description of functions belonging to function spaces with the lattice property. In particular, it is valid for Banach function spaces provided that the Hardy-Littlewood maximal operator is bounded. We also study…

We show that the ultralimit of a bounded sequence of Lipschitz maps into pointed metric spaces extends naturally to $p$-bounded sequences of Sobolev maps and that this ultralimit for Sobolev maps enjoys desirable properties. We use this to…

Let $X$ be a noncomplete metric space satisfying the usual (local) assumptions of a doubling property and a Poincar\'e inequality. We study extensions of Newtonian Sobolev functions to the completion $\widehat{X}$ of $X$ and use them to…

We study the boundary traces of Newton-Sobolev, Hajlasz-Sobolev, and BV (bounded variation) functions. Assuming less regularity of the domain than is usually done in the literature, we show that all of these function classes achieve the…

We establish a new global endpoint Sobolev inequality for measures that extends the classical theorem of Meyers-Ziemer by placing a maximal function on the right-hand side. This result has several significant consequences. It extends…

The aim of this paper is to study ultralimits of pointed metric measure spaces (possibly unbounded and having infinite mass). We prove that ultralimits exist under mild assumptions and are consistent with the pointed measured…

We prove compactness of the embeddings in Sobolev spaces for fractional super and sub harmonic functions with radial symmetry. The main tool is a pointwise decay for radially symmetric functions belonging to a function space defined by…