The Law of the iterated logaritm for smooth functions

TL;DR

This paper proves a version of the Law of the Iterated Logarithm for smooth functions in the upper-half space, revealing growth properties under specific gradient conditions without requiring harmonicity.

Contribution

It introduces a new law of the iterated logarithm for smooth functions and demonstrates self-improvement growth properties based on gradient size conditions.

Findings

Established a Law of the Iterated Logarithm for smooth functions

Identified growth properties linked to gradient and Laplacian gradient conditions

Applied results to non-harmonic contexts

Abstract

A version of the Law of the Iterated Logarithm for smooth functions in the upper-half space is proved. As a consequence, we show that certain size conditions on the gradient and the gradient of the laplacian of a smooth function, lead to self-improvement growth properties. The results are applied in situations where harmonicity is not present.