A Holder Continuous Nowhere Improvable Function with Derivative Singular Distribution

Details A Holder Continuous Nowhere Improvable Function with Derivative Singular Distribution A Holder Continuous Nowhere Improvable Function with Derivative Singular Distribution

2010-11-28

arxiv.org/abs/1011.6071v3

Mathematics

Classical Analysis and ODEs

3 figures, revised result

Scientific paper

We present a class of functions which is a variant of the Knopp class of nowhere differentiable continuous functions. We derive precise estimates, establishing that they are locally H\"older continuous in $C^{0,\al}(\R)$ for any $0<\al<1$ but pointwise nowhere improvable to $C^{0,\be}$ for any better exponent. In particular, they are nowhere differentiable on $\R$ with derivatives singular distributions in $\mD'(\R)$. These functions furnish explicit realizations of the functional analytic result of Berezhnoi \cite{Be} and are introduced in conjunction to regularity theory of nonlinear degenerate 2nd order elliptic partial differential equations and systems, like the $\infty$-Laplacian $\Delta_{\infty}$ and the Aronsson system, allowing to construct pathological solutions.

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