Heli Tuominen

Measure density and extendability of Sobolev functions

We study necessary and sufficient conditions for a domain to be a Sobolev extension domain in the setting of metric measure spaces. In particular, we prove that extension domains must satisfy a measure density condition.

Fractional Hardy inequalities and visibility of the boundary

LIZAVETAIHNATSYEVA,JUHALEHRBACK,HELITUOMINEN,ANDANTTIV.VAHAKANGASAbstract. We prove fractional order Hardy inequalities on open sets under a combined fat-nessandvisibilityconditionontheboundary. Wedemonstratebycounterexamplesthatfatnessconditions alone are not sufficient for such Hardy inequalities to hold. In addition, we give ashort exposition of various fatness conditions related to our main result, and apply fractionalHardyinequalitiesinconnectiontotheboundednessofextensionoperatorsforfractionalSobolevspaces.

Mapping properties of the discrete fractional maximal operator in metric measure spaces

This work studies boundedness properties of the fractional maximal operator on metric measure spaces under standard assumptions on the measure. The main motivation is to show that the fractional maximal operator has similar smoothing and mapping properties as the Riesz potential. Instead of the usual fractional maximal operator, we also consider a so-called discrete maximal operator which has better regularity. We study the boundedness of the discrete fractional maximal operator in Sobolev, Holder, Morrey and Campanato spaces. We also prove a version of the Coifman-Rochberg lemma for the fractional maximal function.

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 a general setting, but they are new already in the Euclidean case.

Fractional maximal functions in metric measure spaces

Abstract 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.

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.

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) = dist(x,E)−α, where E is a closed set in X and α∈ℝ

Self-improvement of uniform fatness revisited

\alpha \in \mathbb {R}

Sobolev embeddings, extensions and measure density condition

. We establish sharp conditions, based on the Assouad (co)dimension of E, for the inclusion of w in Muckenhoupt’s Ap classes of weights, 1 ≤ p < ∞. With the help of general Ap-weighted embedding results, we then prove (global) Hardy–Sobolev inequalities and also fractional versions of such inequalities in the setting of metric spaces.

Lebesgue points and capacities via the boxing inequality in metric spaces

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.