Searcharxiv⌕ Search

arXiv subjects

Heli Tuominen

Publications and source records attributed to Heli Tuominen.

12 recordsLinked to original sources
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↗
arXiv2015

Self-improvement of uniform fatness revisited

We give a new proof for the self-improvement of uniform p-fatness in the setting of general metric spaces. Our proof is based on rather standard methods of geometric analysis, and in particular the proof avoids the use of deep results from potential theory and analysis on metric spaces that have been indispensable in the previous proofs of the self-improvement. A key ingredient in the proof is a self-improvement property for local Hardy inequalities.

math.CA↗
arXiv2015

Removable singularities for div v = f in weighted Lebesgue spaces

Let $w\in L^1\_{loc}(\R^n)$ be apositive weight. Assuming that a doubling condition and an $L^1$ Poincaré inequality on balls for the measure $w(x)dx$, as well as a growth condition on $w$, we prove that the compact subsets of $\R^n$ which are removable for the distributional divergence in $L^{\infty}\_{1/w}$ are exactly those with vanishing weighted Hausdorff measure. We also give such a characterization for $L^p\_{1/w}$, $1\textless{}p\textless{}+\infty$, in terms of capacity. This generalizes results due to Phuc and Torres, Silhavy and the first author.

math.CA↗
arXiv2015

Approximation and quasicontinuity of Besov and Triebel-Lizorkin functions

We show that, for $0<s<1$, $0<p<\infty$, $0<q<\infty$, Hajłasz-Besov and Hajłasz-Triebel-Lizorkin functions can be approximated in the norm by discrete median convolutions. This allows us to show that, for these functions, the limit of medians, \[ \lim_{r\to 0}m_u^γ(B(x,r))=u^*(x), \] exists quasieverywhere and defines a quasicontinuous representative of $u$. The above limit exists quasieverywhere also for Hajłasz functions $u\in M^{s,p}$, $0<s\le 1$, $0<p<\infty$, but approximation of $u$ in $M^{s,p}$ by discrete (median) convolutions is not in general possible.

math.FA↗
arXiv2015

Approximation by Hölder functions in Besov and Triebel-Lizorkin spaces

In this paper, we show that Besov and Triebel-Lizorkin functions can be approximated by a Hölder continuous function both in the Lusin sense and in norm. The results are proven in metric measure spaces for Hajłasz-Besov and Hajłasz-Triebel-Lizorkin functions defined by a pointwise inequality. We also prove new inequalities for medians, including a Poincaré type inequality, which we use in the proof of the main result.

math.FA↗
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

Regularity of the local fractional maximal function

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 norm estimates in Sobolev spaces. An unexpected feature is that these estimates contain extra terms involving spherical and fractional maximal functions. Moreover, we construct several explicit examples which show that our results are essentially optimal. Extensions to metric measure spaces are also discussed.

math.FA↗
arXiv2013

Fractional maximal functions in metric measure spaces

We study the mapping properties of fractional maximal operators in Sobolev and Campanato spaces in metric measure spaces. We show that, under certain restrictions on the underlying metric measure space, fractional maximal operators improve the Sobolev regularity of functions and map functions in Campanato spaces to Hölder continuous functions. We also give an example of a space where fractional maximal function of a Lipschitz function fails to be continuous.

math.FA↗
arXiv2013

Smoothing properties of the discrete fractional maximal operator on Besov and Triebel--Lizorkin spaces

Motivated by the results of Korry and Kinnunen and Saksman, we study the behaviour of the discrete fractional maximal operator on fractional Hajlasz spaces, Hajlasz-Besov and Hajlasz-Triebel-Lizorkin spaces on metric measure spaces. We show that the discrete fractional maximal operator maps these spaces to the spaces of the same type with higher smoothness. Our results extend and unify aforementioned results. We present our results in general setting, but they are new already in the Euclidean case.

math.FA↗
arXiv2009

The equivalence between pointwise Hardy inequalities and uniform fatness

We prove an equivalence result between the validity of a pointwise Hardy inequality in a domain and uniform capacity density of the complement. This result is new even in Euclidean spaces, but our methods apply in general metric spaces as well. We also present a new transparent proof for the fact that uniform capacity density implies the classical integral version of the Hardy inequality in the setting of metric spaces. In addition, we consider the relations between the above concepts and certain Hausdorff content conditions.

math.AP↗