Generalized Differentiability of Continuous Functions

Many physical phenomena give rise to mathematical models in terms of fractal, non-differentiable functions. The paper introduces a broad generalization of the derivative in terms of the maximal modulus of continuity of the primitive function. These derivatives are called indicial derivatives. As an application, the indicial derivatives are used to characterize the nowhere monotonous functions. Furthermore, the non-differentiability set of such derivatives is proven to be of measure zero. As a second application, the indicial derivative is used in the proof of the Lebesgue differentiation theorem. Finally, the connection with the fractional velocities is demonstrated.

Besicovitch Covering Property on graded groups and applications to measure differentiation

We give a complete answer to which homogeneous groups admit homogeneous distances for which the Besicovitch Covering Property (BCP) holds. In particular, we prove that a stratified group admits homogeneous distances for which BCP holds if and only if the group has step 1 or 2. These results are obtained as consequences of a more general study of homogeneous quasi-distances on graded groups. Namely, we prove that a positively graded group admits continuous homogeneous quasi-distances satisfying BCP if and only if any two different layers of the associated positive grading of its Lie algebra commute. The validity of BCP has several consequences. Its connections with the theory of differentiation of measures is one of the main motivations of the present paper. As a consequence of our results, we get for instance that a stratified group can be equipped with some homogeneous distance so that the differentiation theorem holds for each locally finite Borel measure if and only if the group has step 1 or 2. The techniques developed in this paper allow also us to prove that sub-Riemannian distances on stratified groups of step 2 or higher never satisfy BCP. Using blow-up techniques this is shown to imply that on a sub-Riemannian manifold the differentiation theorem does not hold for some locally finite Borel measure.

Norms supporting the Lebesgue differentiation theorem

A version of the Lebesgue differentiation theorem is offered, where the [Formula: see text] 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.

A FUNDAMENTAL DICHOTOMY FOR DEFINABLY COMPLETE EXPANSIONS OF ORDERED FIELDS

An expansion of a definably complete field either defines a discrete subring, or the image of every definable discrete set under every definable map is nowhere dense. As an application we show a definable version of Lebesgue’s differentiation theorem.