Differential calculus on topological spaces with weak Markov structure. I
The motivation of the paper comes from the Whitney problems for differentiable functions. It considers a generalization of differential calculus to more general settings than Euclidean spaces and manifolds. It focuses on (possibly fractal) subsets of \(\mathbb{R}^n\) (and a corresponding generalization of manifolds). More precisely, the studied sets are called \textit{weak \(k\)-Markov} for some \(k\in \mathbb{N}_0\). Before talking about weak Markov sets, we need to tell when a point is weak Markov; we let \(\mathcal{P}_{k,n}\) be the space of real polynomials on \(\mathbb{R}^n\) of degree \(k\) and \(Q_r(x)\subset \mathbb{R}^n\) be the closed cube centered at \(x\) of side length \(2r\). Given a subset \(S\subset \mathbb{R}\), a point \(x\in S\) is said to be \textit{weak \(k\)-Markov} if \[ \varliminf_{r\to 0} \left\{\sup_{p\in \mathcal{P}_{k,n}\setminus\{0\}}\left(\frac{\sup_{Q_r(x)}\left|p\right|}{\sup_{S\cap Q_r(x)}\left|p\right|}\right)\right\}<\infty. \] A closed set \(S\subset \mathbb{R}\) is said to be \textit{k-Markov} if it contains a dense subset of weak \(k\)-Markov points. The term \textit{weak Markov set} is a reference to \textit{Markov set} (see [\textit{A. Jonsson} and \textit{H. Wallin}, Function spaces on subsets of \(\text{R}^ n\). Math. Rep. Ser. 2, No. 1 (1984; Zbl 0875.46003)]). After introducing weak Markov sets, a section is dedicated to some of their properties. Then traces of differentiable functions to weak Markov sets are studied. Next, the focus shifts to extension problems. For example, a Hartogs-type problem is solved and also certain harmonic functions constructed. The next section leaves the Euclidean setting by introducing \textit{weak Markov structures}. They are reminiscent of manifolds. Some properties are stated and a calculus of differential forms established. Generalizations of some classical theorems (the Poincaré \(d\)-lemma for differential forms on \(C^\infty\) manifolds, the de Rham theorem, and the Künneth formula) are obtained. The remaining sections deal with the proofs of the statements in the first part of the paper.
scientific article; zbMATH DE number 6496472
scientific article