A Controlled Hahn-Mazurkiewicz Theorem and its Applications

For a metric Peano continuum $X$, let $S_X$ be a Sierpiński function assigning to each $\epsilon >0$ the smallest cardinality of a cover of $X$ by connected subsets of diameter $\le \epsilon$. We prove that for any increasing function $\mathrm{\Omega }:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ with $(0,1]\subseteq \mathrm{\Omega }[{\mathbb{R}}_{+}]$ and $s:={\sum }_{n=1}^{\mathrm{\infty }}{S}_X(2^{-n}){\sum }_{m=n}^{\mathrm{\infty }}{S}_X(2^{-m}){\mathrm{\Omega }}^{-1}(\min \{1,2^{6-m}\})<\mathrm{\infty }$ there exists a continuous surjective function $f:[0,s]\to X$ with continuity modulus ${\omega }_f\le \mathrm{\Omega }$. This controlled version of the classical Hahn-Mazurkiewicz Theorem implies that $SDim(X)\le HDim(X)\le 2\cdot SDim(X)$, where $SDim(X)={\mathrm{lim\hspace{0.17em}sup}}_{\epsilon \to 0}\frac{\ln (S_X(\epsilon ))}{\ln (1\mathrm{/}\epsilon )}$ is the $S$-dimension of $X$, and $HDim(X)=\inf \{\alpha \in (0,\mathrm{\infty }]:$ there is a~surjective $\frac{1}{\alpha }$-Hölder map $f:[0,1]\to X\}$ is the $H\ddot{o}lder$ $dimension$ of $X$.