Analogues of $s$-potential and $s$-energy of mass distribution on Cantor dyadic group and their relation to Hausdorff dimension

Abstract

We introduce an analogue of Riesz $s$-potetial and $s$-energy, $0<s<1$, of a mass distribution $\mu$ on the Cantor dyadic group $G$ by defining a respactive $s$-kernel. Then we relate Hausdorff dimension of a set $E\subset G$ to the value of $s$-energy of the mass distribution $\mu$ on this set $E$. Namely we prove that if on a set $E$ there exists a mass distribution $\mu$ with finite $s$-energy, then the Hausdorff dimension of $E$ is at least $s$. The same condiion can be expressed also in terms of Fourier coefficients of $\mu$ with respect to Walsh system on the group $G$.