arXiv subjects

Michael Dymond

Publications and source records attributed to Michael Dymond.

21 records · Page 2Linked to original sources

arXiv2014

On interval based generalizations of absolute continuity for functions on $\mathbb{R}^n$

We study notions of absolute continuity for functions defined on $\mathbb{R}^n$similar to the notion of $α$-absolute continuity in the sense of Bongiorno. We confirm a conjecture of Malý that 1-absolutely continuous functions do not need to be differentiable a.e., and we show several other pathological examples of functions in this class. We establish containment relations of the class $1-AC_{\rm WDN}$ which consits of all functions in $1-AC$ which are in the Sobolev space $W^{1,2}_{loc}$, are differentiable a.e. and satisfy the Luzin (N) property, with previously studied classes of absolutely continuous functions.

math.FA↗

arXiv2013

Avoiding σ-porous sets in Hilbert spaces

We give a constructive proof that any $σ$-porous subset of a Hilbert space has Lebesgue measure zero on typical $C^{1}$ curves. Further, we discover that this result does not extend to all forms of porosity; we find that even power-$p$ porous sets may meet many $C^{1}$ curves in positive measure.

math.FA↗

arXiv2013

Differentiability inside sets with upper Minkowski dimension one

We show that every finite-dimensional Euclidean space contains compact universal differentiability sets of upper Minkowski dimension one. In other words, there are compact sets $S$ of upper Minkowski dimension one such that every Lipschitz function defined on the whole space is differentiable inside $S$. Such sets are constructed explicitly.

math.FA↗