Antti V. Vähäkangas

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.

On the Local Tb Theorem: A Direct Proof under the Duality Assumption

We give a new direct proof of the local Tb theorem in the Euclidean setting and under the assumption of dual exponents. This theorem provides a flexible framework for proving the boundedness of a Calderon–Zygmund operator, supposing the existence of systems of local accretive functions. We assume that the integrability exponents on these systems of functions are of the form 1 /p + 1 /q ⩽ 1, the ‘dual case’ 1 /p + 1 /q = 1 being the most difficult one. Our proof is direct: it avoids a reduction to the perfect dyadic case unlike some previous approaches. The principal point of interest is in the use of random grids and the corresponding construction of the corona. We also use certain twisted martingale transform inequalities.

Aspects of local-to-global results

We establish local-to-global results for a function space which is larger than the well-known bounded mean oscillation space, and was also introduced by John and Nirenberg

In between the inequalities of Sobolev and Hardy

We establish both sufficient and necessary conditions for the validity of the so-called Hardy–Sobolev inequalities on open sets of the Euclidean space. These inequalities form a natural interpolating scale between the (weighted) Sobolev inequalities and the (weighted) Hardy inequalities. The Assouad dimension of the complement of the open set turns out to play an important role in both sufficient and necessary conditions.

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}

On fractional Poincaré inequalities

. 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.

A FRAMEWORK FOR FRACTIONAL HARDY INEQUALITIES

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.