Estimates for potentials and \(\delta\)-subharmonic functions outside exceptional sets.

*(English. Russian original)*Zbl 0904.31003
Izv. Math. 61, No. 6, 1293-1329 (1997); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 61, No. 6, 181-218 (1997).

Let \(\mu\) be a non-negative measure on \(\mathbb{R}^m\) \((m\geq 1)\) and let \(K: (0,+\infty)\to [0,+\infty)\) be continuous and decreasing. Let \(U\) be defined on \(\mathbb{R}^m\) by the equation \(U(x)= \int K(| x-y|) d\mu(y)\). Let \(B(x,r)\) denote the open ball of centre \(x\) and radius \(r\) in \(\mathbb{R}^m\). The following theorem is proved by using established techniques. Theorem A: Assume that \(\mu(\mathbb{R}^m)= N<+\infty\), and let \(h\) be an increasing function on \((0,r_0)\) with limiting values \(h(0)= 0\) and \(h(r_0)= N\). Then the set \(G\) of points \(x\) for which \(U(x)> \int_0^{r_0} K(t)dh(t)\) can be covered by countably many balls \(B(x_k,r_k)\) for which \(r_k\leq r_0\) and \(\sum_k h(r_k)< A_m N\), where \(A_m\) depends only on \(m\). The multiplicity of the covering does not exceed \(A_m\), and \(h(r_k)< \mu(B(x_k,r_k))\) for each \(k\). In the case where \(K\) is a power function or a logarithmic function, Theorem A is due to N. S. Landkof.

In the paper under review it is shown that the estimate of the size of the set \(G\) in Theorem A is, in a sense, exact. This is achieved by establishing estimates for the Hausdorff measure and capacity of Cantor sets in \(\mathbb{R}^m\) and estimates for potentials on these sets. These results are also used to improve and extend other known theorems. For example, Frostman’s theorem on the comparison between Hausdorff measure and capacity is supplemented by inequalities connecting capacity and \(h\)-girth in the sense of Hausdorff. Also an exact condition on the measuring function \(h\) is found under which the convergence of \(\int_0 K(t)h(t)dt\) is necessary for the validity of Frostman’s theorem. The final section of the paper gives estimates for certain \(\delta\)-subharmonic functions on a ball outside exceptional sets. These results generalize a theorem of Govorov for subharmonic functions on a disc, which itself extends the Valiron-Bernstein theorem on the lower estimation of the modulus of a holomorphic function.

In the paper under review it is shown that the estimate of the size of the set \(G\) in Theorem A is, in a sense, exact. This is achieved by establishing estimates for the Hausdorff measure and capacity of Cantor sets in \(\mathbb{R}^m\) and estimates for potentials on these sets. These results are also used to improve and extend other known theorems. For example, Frostman’s theorem on the comparison between Hausdorff measure and capacity is supplemented by inequalities connecting capacity and \(h\)-girth in the sense of Hausdorff. Also an exact condition on the measuring function \(h\) is found under which the convergence of \(\int_0 K(t)h(t)dt\) is necessary for the validity of Frostman’s theorem. The final section of the paper gives estimates for certain \(\delta\)-subharmonic functions on a ball outside exceptional sets. These results generalize a theorem of Govorov for subharmonic functions on a disc, which itself extends the Valiron-Bernstein theorem on the lower estimation of the modulus of a holomorphic function.

Reviewer: D.H.Armitage (Belfast)

##### MSC:

31B05 | Harmonic, subharmonic, superharmonic functions in higher dimensions |

31B15 | Potentials and capacities, extremal length and related notions in higher dimensions |