Integrals of differences of subharmonic functions. III. Hausdorff measure and content, integration over Lipschitz curves and surfaces

TL;DR

This paper develops new integral inequalities for differences of subharmonic functions using Borel measures, extending to fractal supports and providing estimates via Hausdorff measures, with applications to complex analysis and geometric measure theory.

Contribution

It introduces novel integral inequalities involving subharmonic functions, Borel measures, and Hausdorff measures, applicable to fractal supports and various geometric configurations.

Findings

New integral inequalities for subharmonic functions are established.

Estimates are extended to fractal supports using Hausdorff measure and content.

Results apply to complex functions, Lipschitz curves, and hypersurfaces.

Abstract

Additional integral inequalities are obtained for integrals of the differences of subharmonic functions by Borel measures on balls in a multidimensional Euclidean space. These integrals are still estimated from above through the Nevanlinna characteristic and various characteristics of the Borel measure and its support. All the results are also new for logarithms of modules of meromorphic functions on discs in the complex plane. Integration by Borel measures with a support on fractal sets is allowed, and estimates in these cases are given by the Hausdorff measure and the Hausdorff content for the support of the Borel measure. We consider separately important for applications special cases of functions in the entire complex plane and space, in the unit disc or ball, as well as cases of integration over length on subsets of Lipschitz curves and over area on subsets of Lipschitz hypersurfaces.