Estimates for potentials and $\delta$-subharmonic functions outside exceptional sets

V. Ya. Èiderman

Izvestiya. Mathematics , Tome 61 (1997) no. 6, pp. 1293-1329

Résumé

It is shown that the estimates for potentials obtained by Landkof are, in a sense, unimprovable. To prove this, we establish exact estimates for the Hausdorff measure and the capacity of Cantor sets in $\mathbb R^m$, $m\geqslant 1$, and estimates for potentials on these sets. These results are used in other sections of this article. Frostman's theorem on the comparison of the Hausdorff measure with the capacity is supplemented with inequalities that connect the capacity and the $h$-girth in the sense of Hausdorff. We find an exact condition on measuring functions under which convergence of the integral $\int_0K(t)\,dh(t)$ is necessary for the validity of Frostman's theorem (here $h$ is the measuring function and $K$ is the kernel of the potential). The theorem of Govorov on the estimation of a subharmonic function in a disc (which, in turn, extends the Valiron–Bernstein theorem on the lower estimation of the modulus of a holomorphic function) is generalized to $\delta$-subharmonic functions of bounded form in a ball in $\mathbb R^m$, $m\geqslant 2$. In connection with this, we consider a more general problem rather than the problem of estimating the sum of the radii of exceptional balls. We study the exactness of the results obtained.