Maximal function estimates and self-improvement results for Poincaré inequalities

Our main result is an estimate for a sharp maximal function, which implies a Keith–Zhong type self-improvement property of Poincaré inequalities related to differentiable structures on metric measure spaces. As an application, we give structure independent representation for Sobolev norms and universality results for Sobolev spaces. peerReviewed

Perron's method for the porous medium equation

O. Perron introduced his celebrated method for the Dirichlet problem for harmonic functions in 1923. The method produces two solution candidates for given boundary values, an upper solution and a lower solution. A central issue is then to determine when the two solutions are actually the same function. The classical result in this direction is Wiener’s resolutivity theorem: the upper and lower solutions coincide for all continuous boundary values. We discuss the resolutivity theorem and the related notions for the porous medium equation ut −∆u = 0

The De Giorgi measure and an obstacle problem related to minimal surfaces in metric spaces

Abstract We study the existence of a set with minimal perimeter that separates two disjoint sets in a metric measure space equipped with a doubling measure and supporting a Poincare inequality. A measure constructed by De Giorgi is used to state a relaxed problem, whose solution coincides with the solution to the original problem for measure theoretically thick sets. Moreover, we study properties of the De Giorgi measure on metric measure spaces and show that it is comparable to the Hausdorff measure of codimension one. We also explore the relationship between the De Giorgi measure and the variational capacity of order one. The theory of functions of bounded variation on metric spaces is us…

A maximal Function Approach to Two-Measure Poincaré Inequalities

This paper extends the self-improvement result of Keith and Zhong in  Keith and Zhong (Ann. Math. 167(2):575–599, 2008) to the two-measure case. Our main result shows that a two-measure (p, p)-Poincare inequality for $$10$$ under a balance condition on the measures. The corresponding result for a maximal Poincare inequality is also considered. In this case the left-hand side in the Poincare inequality is replaced with an integral of a sharp maximal function and the results hold without a balance condition. Moreover, validity of maximal Poincare inequalities is used to characterize the self-improvement of two-measure Poincare inequalities. Examples are constructed to illustrate the role of t…

Maximal Function Methods for Sobolev Spaces

REGULARITY OF THE FRACTIONAL MAXIMAL FUNCTION

The purpose of this work is to show that the fractional maximal operator has somewhat unexpected regularity properties. The main result shows that the fractional maximal operator maps -spaces boundedly into certain first-order Sobolev spaces. It is also proved that the fractional maximal operator preserves first-order Sobolev spaces. This extends known results for the Hardy–Littlewood maximal operator.

Higher Order Sobolev-Type Spaces on the Real Line

This paper gives a characterization of Sobolev functions on the real line by means of pointwise inequalities involving finite differences. This is also shown to apply to more general Orlicz-Sobolev, Lorentz-Sobolev, and Lorentz-Karamata-Sobolev spaces.