arXiv subjects
Publications and source records attributed to Jakub Duda.
The notion of Aronszajn-null sets generalizes the notion of Lebesgue measure zero on the real line and the Euclidean space to infinite dimensional Banach spaces. We present a game-theoretic approach to Aronszajn-null sets, establish its basic properties, and discuss the ensuing open problems.
We give a complete characterization of those $f: [0,1] \to X$ (where $X$ is a Banach space which admits an equivalent Fréchet smooth norm) which allow an equivalent $C^2$ parametrization. For $X=\R$, a characterization is well-known. However, even in the case $X=\R^2$, several quite new ideas are needed. Moreover, the very close case of parametrizations with a bounded second derivative is solved.
We find an equivalent condition for a real function $f:[a,b]\to\R$ to be Lebesgue equivalent to an $n$-times differentiable function ($n\geq 2$); a simple solution in the case $n=2$ appeared in an earlier paper. For that purpose, we introduce the notions of $CBVG_{1/n}$ and $SBVG_{1/n}$ functions, which play analogous roles for the $n$-th order differentiability as the classical notion of a $VBG_*$ function for the first order differentiability, and the classes $CBV_{1/n}$ and $SBV_{{1}/{n}}$ (introduced by Preiss and Laczkovich) for $C^n$ smoothness. As a consequence of our approach, we obtain that Lebesgue equivalence to $n$-times differentiable function is the same as Lebesgue equivalence to a function $f$ which is $(n-1)$-times differentiable with $f^{(n-1)}(\cdot)$ being pointwise Lipschitz. We also characterize the situation when a given function is Lebesgue equivalent to an $n$-times differentiable function $g$ such that $g'$ is nonzero a.e. As a corollary, we establish a generalization of Zahorski's Lemma for higher order differentiability.
We define the notions of unilateral metric derivatives and ``metric derived numbers'' in analogy with Dini derivatives (also referred to as ``derived numbers'') and establish their basic properties. We also prove that the set of points where a path with values in a metric space with continuous metric derivative is not ``metrically differentiable'' (in a certain strong sense) is $σ$-symmetrically porous and provide an example of a path for which this set is uncountable. In the second part of this paper, we study the continuous metric differentiability via a homeomorphic change of variable.
We prove that for every function $f:X\to Y$, where $X$ is a separable Banach space and $Y$ is a Banach space with RNP, there exists a set $A\in\tilde\mcA$ such that $f$ is Gateaux differentiable at all $x\in S(f)\setminus A$, where $S(f)$ is the set of points where $f$ is pointwise-Lipschitz. This improves a result of Bongiorno. As a corollary, we obtain that every $K$-monotone function on a separable Banach space is Hadamard differentiable outside of a set belonging to $\tilde\mcC$; this improves a result due to Borwein and Wang. Another corollary is that if $X$ is Asplund, $f:X\to\R$ cone monotone, $g:X\to\R$ continuous convex, then there exists a point in $X$, where $f$ is Hadamard differentiable and $g$ is Frechet differentiable.
We find an equivalent condition for a continuous vector-valued path to be Lebesgue equivalent to a twice differentiable function. For that purpose, we introduce the notion of a $VBG_{1/2}$ function, which plays an analogous role for the second order differentiability as the classical notion of a $VBG_*$ function for the first order differentiability. In fact, for a function $f:[a,b]\to X$, being Lebesgue equivalent to a twice differentiable function is the same as being Lebesgue equivalent to a differentiable function with a pointwise Lipschitz derivative. We also consider the case when the first derivative can be taken non-zero almost everywhere.
We generalize some results of Borwein, Burke, Lewis, and Wang to mappings with values in metric (resp. ordered normed linear) spaces. We define two classes of monotone mappings between an ordered linear space and a metric space (resp. ordered linear space): $K$-monotone dominated and cone-to-cone monotone mappings. First we show some relationships between these classes. Then, we study continuity and differentiability (also in the metric and $w^*$ senses) of mappings in these classes.
We present a general theory of absolutely continuous paths with values in metric spaces using the notion of metric derivatives. Among other results, we prove analogues of the Banach-Zarecki and Vallee Poussin theorems.