Related papers: A note on Lusin-type approximation of Sobolev func…

We establish new approximation results, in the sense of Lusin, of Sobolev functions by Lipschitz ones, in some classes of non-doubling metric measure structures. Our proof technique relies upon estimates for heat semigroups and applies to…

We discuss new proofs, and new forms, of a reverse logarithmic Sobolev inequality, with respect to the standard Gaussian measure, for low complexity functions, measured in terms of Gaussian-width. In particular, we provide a dimension-free…

Functions in a Sobolev space are approximated directly by piecewise affine interpolation in the norm of the space. The proof is based on estimates for interpolations and does not rely on the density of smooth functions.

In this paper, we show that Besov and Triebel-Lizorkin functions can be approximated by a H\"older continuous function both in the Lusin sense and in norm. The results are proven in metric measure spaces for Haj{\l}asz-Besov and…

Let $X$ be a separable Banach space endowed with a non-degenerate centered Gaussian measure $\mu$ and let $w$ be a positive function on $X$ such that $w\in W^{1,s}(X,\mu)$ and $\log w\in W^{1,t}(X,\mu)$ for some $s>1$ and $t>s'$. In the…

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

A new representation is proposed for functions in a Sobolev space with dominating mixed smoothness on an $N$-dimensional hyperrectangle. In particular, it is shown that these functions can be expressed in terms of their highest-order mixed…

We prove a Lusin approximation of functions of bounded variation. If $f$ is a function of bounded variation on an open set $\Omega\subset X$, where $X=(X,d,\mu)$ is a given complete doubling metric measure space supporting a $1$-Poincar\'e…

We study Lusin-measurable functions with values in locally convex spaces. In particular, the behavior of pointwise limits of sequences of Lusin-measurable functions and exhibit pathological phenomena arising in the nonmetrizable setting.…

In this paper we show a density property for fractional weighted Sobolev spaces. That is, we prove that any function in a fractional weighted Sobolev space can be approximated by a smooth function with compact support. The additional…

This paper introduces first order Sobolev spaces on certain rectifiable varifolds. These complete locally convex spaces are contained in the generally nonlinear class of generalised weakly differentiable functions and share key functional…

We study the Lusin approximation problem for real-valued measurable functions on Carnot groups. We prove that k-approximate differentiability almost everywhere is equivalent to admitting a Lusin approximation by $C^{k}_{\mathbb{G}}$ maps.…

We deduce an extension theorem for the so-called Sobolev-Grand Lebesgue Spaces defined on the suitable subsets of the whole finite-dimensional Euclidean space, and estimate the norms of correspondent extension operator, which may be choosed…

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…

In a 2013 paper, the author showed that the convolution of a compactly supported measure on the real line with a Gaussian measure satisfies a logarithmic Sobolev inequality (LSI). In a 2014 paper, the author gave bounds for the optimal…

We study extensions of Sobolev and BV functions on infinite-dimensional domains. Along with some positive results we present a negative solution of the long-standing problem of existence of Sobolev extensions of functions in Gaussian…

For a Lipschitz function $f$ on an open set in $\mathbb{R}^n$, we consider the $L^{n}$ integrability of the quotient $\frac{|\nabla f|}{|f|}$ over its natural domain of definition.

We prove an approximation lemma on (stratified) homogeneous groups that allows one to approximate a function in the non-isotropic Sobolev space $\dot{NL}^{1,Q}$ by $L^{\infty}$ functions, generalizing a result of Bourgain-Brezis…

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…

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…