We study a function space JNp based on a condition introduced by John and Nirenberg as a variant of BMO. It is known that L p ⊂ JNp L p,∞ , but otherwise the structure of JNp is largely a mystery. Our first main result is the construction of a function that belongs to JNp but not L p , showing that the two spaces are not the same. Nevertheless, we prove that for monotone functions, the classes JNp and L p do coincide. Our second main result describes JNp as the dual of a new Hardy kind of space HK p ′ .

Abstract. The purpose of this work is to study regularity of Sobolev functions on metric measure spaces equipped with a doubling measure and supporting a weak Poincaré inequality. We show that every Sobolev function whose gradient is integrable to power one has Lebesgue points outside a set of 1-capacity zero. We also show that 1-capacity is equivalent to the Hausdorff content of codimension one and study characterizations of 1-capacity in terms of Frostman's lemma and functions of bounded variation. As the main technical tool, we prove a metric space version of Gustin's boxing inequality. Our proofs are based on covering arguments and functions of bounded variation. Perimeter measures, isoperimetric inequalities and coarea formula play an essential role in our approach.

Abstract. We show to a general class of parabolic equations that every bounded superparabolic function is a weak supersolution and, in particular, has derivatives in a Sobolev sense. To this end, we establish various comparison principles between super-and subparabolic functions, and show that a pointwise limit of uniformly bounded weak supersolutions is a weak supersolution.

In this paper we establish that the gradient of weak solutions to porous medium-type systems admits the self-improving property of higher integrability.

We study pointwise properties of functions of bounded variation on a metric space equipped with a doubling measure and a Poincaré inequality. In particular, we obtain a Lebesgue type result for BV functions. We also study approximations by Lipschitz continuous functions and a version of the Leibniz rule. We give examples which show that our main result is optimal for BV functions in this generality.