Narrow Results

A version of the Lebesgue differentiation theorem is offered, where the $L^p$ norm is replaced with any rearrangement-invariant norm. Necessary and sufficient conditions for a norm of this kind to support the Lebesgue differentiation theorem are established. In particular, Lorentz, Orlicz and other customary norms for which Lebesgue’s theorem holds are characterized.

We study compactness of embeddings of Sobolev-type spaces of arbitrary integer order into function spaces on domains in ${ℝ}^n$ with respect to upper Ahlfors regular measures $\nu$, that is, Borel measures whose decay on balls is dominated by a power of their radius. Sobolev-type spaces as well as target spaces considered in this paper are built upon general rearrangement-invariant function norms. Several sufficient conditions for compactness are provided and these conditions are shown to be often also necessary, yielding sharp compactness results. It is noteworthy that the only connection between the measure $\nu$ and the compactness criteria is how fast the measure decays on balls. Applications to Sobolev-type spaces built upon Lorentz–Zygmund norms are also presented.

We survey some recent works ([3,11,13]) concerning the study of the norm of the Hardy operator minus the Identity. In particular, optimal estimates are found for the cone of radially decreasing functions on the minimal Lorentz spaces Λ(X) (restricted type estimates), and a new Banach function space, closely related to Λ(X), is also naturally defined. We also mention a recent solution to a conjecture of Kruglyak and Setterqvist.