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Lizaveta Ihnatsyeva

Publications and source records attributed to Lizaveta Ihnatsyeva.

13 recordsLinked to original sources
arXiv2024

Fractional Hardy inequalities and capacity density

We prove that a pointwise fractional Hardy inequality implies a fractional Hardy inequality, defined via a Gagliardo-type seminorm. The proof consists of two main parts. The first one is to characterize the pointwise fractional Hardy inequality in terms of a fractional capacity density condition. The second part is to show the deep open-endedness or self-improvement property of the fractional capacity density, which we accomplish in the setting of a complete geodesic space equipped with a doubling measure. These results are new already in the standard Euclidean setting.

math.CA↗
arXiv2022

Capacities and density conditions in metric spaces

We examine the relations between different capacities in the setting of a metric measure space. First, we prove a comparability result for the Riesz $(\beta,p)$-capacity and the relative Hajlasz $(\beta,p)$-capacity, for $1<p<\infty$ and $0<\beta \le 1$, under a suitable kernel estimate related to the Riesz potential. Then we show that in geodesic spaces the corresponding capacity density conditions are equivalent even without assuming the kernel estimate. In the last part of the paper, we compare the relative Hajlasz $(1,p)$-capacity to the relative variational $p$-capacity.

math.AP↗
arXiv2021

Hardy-Sobolev inequalities and weighted capacities in metric spaces

Let $\Omega$ be an open set in a metric measure space $X$. Our main result gives an equivalence between the validity of a weighted Hardy-Sobolev inequality in $\Omega$ and quasiadditivity of a weighted capacity with respect to Whitney covers of $\Omega$. Important ingredients in the proof include the use of a discrete convolution as a capacity test function and a Maz'ya type characterization of weighted Hardy-Sobolev inequalities.

math.CA↗
arXiv2017

Muckenhoupt $A_p$-properties of distance functions and applications to Hardy-Sobolev -type inequalities

Let $X$ be a metric space equipped with a doubling measure. We consider weights $w(x)=\operatorname{dist}(x,E)^{-α}$, where $E$ is a closed set in $X$ and $α\in\mathbb R$. We establish sharp conditions, based on the Assouad (co)dimension of $E$, for the inclusion of $w$ in Muckenhoupt's $A_p$ classes of weights, $1\le p<\infty$. With the help of general $A_p$-weighted embedding results, we then prove (global) Hardy-Sobolev inequalities and also fractional versions of such inequalities in the setting of metric spaces.

math.CA↗
arXiv2014

Fractional Hardy inequalities and visibility of the boundary

We prove fractional order Hardy inequalities on open sets under a combined fatness and visibility condition on the boundary. We demonstrate by counterexamples that fatness conditions alone are not sufficient for such Hardy inequalities to hold. In addition, we give a short exposition of various fatness conditions related to our main result, and apply fractional Hardy inequalities in connection to the boundedness of extension operators for fractional Sobolev spaces.

math.CA↗
arXiv2014

Measure density and extension of Besov and Triebel-Lizorkin functions

We show that a domain is an extension domain for a Hajłasz-Besov or for a Hajłasz-Triebel-Lizorkin space if and only if it satisfies a measure density condition. We use a modification of the Whitney extension where integral averages are replaced by median values, which allows us to handle also the case $0<p<1$. The necessity of the measure density condition is derived from embedding theorems; in the case of Hajłasz-Besov spaces we apply an optimal Lorentz-type Sobolev embedding theorem which we prove using a new interpolation result. This interpolation theorem says that Hajłasz-Besov spaces are intermediate spaces between $L^p$ and Hajłasz-Sobolev spaces. Our results are proved in the setting of a metric measure space, but most of them are new even in the Euclidean setting, for instance, we obtain a characterization of extension domains for classical Besov spaces $B^s_{p,q}$, $0<s<1$, $0<p<\infty$, $0<q\le\infty$, defined via the $L^p$-modulus of smoothness of a function.

math.FA↗
arXiv2013

On improved fractional Sobolev-Poincaré inequalities

We prove a certain improved fractional Sobolev-Poincaré inequality on John domains; the proof is based on the equivalence of the corresponding weak and strong type inequalities. We also give necessary conditions for the validity of an improved fractional Sobolev-Poincaré inequality, in particular, we show that a domain having a finite measure and satisfying this inequality, and a `separation property', is a John domain.

math.CA↗
arXiv2012

Hardy inequalities in Triebel-Lizorkin spaces

We prove an inequality of Hardy type for functions in Triebel-Lizorkin spaces. The distance involved is being measured to a given Ahlfors d-regular set in R^n, with n-1<d<n. As an application of the Hardy inequality, we consider boundedness of pointwise multiplication operators, and extension problems.

math.CA↗
arXiv2012

On Whitney-type characterization of approximate differentiability on metric measure spaces

We study approximately differentiable functions on metric measure spaces admitting a Cheeger differentiable structure. The main result is a Whitney-type characterization of approximately differentiable functions in this setting. As an application, we prove a Stepanov-type theorem and consider approximate differentiability of Sobolev, BV and maximal functions.

math.CA↗
arXiv2011

How to recognize polynomials in higher order Sobolev spaces

This paper extends characterizations of Sobolev spaces by Bourgain, Brézis, and Mironescu to the higher order case. As a byproduct, we obtain an integral condition for the Taylor remainder term, which implies that the function is a polynomial. Similar questions are also considered in the context of Whitney jets.

math.CA↗