"Graph Paper" Trace Characterizations of Functions of Finite Energy

TL;DR

This paper characterizes finite energy functions in the plane and on fractals using their traces on grid lines and Sobolev norms, providing new insights into their structure and properties.

Contribution

It introduces a novel trace characterization of finite energy functions on the plane and fractals, linking geometric structures with Sobolev space norms.

Findings

Finite energy functions are characterized by their traces on grid lines.

Analogous trace characterizations are established for functions on Sierpinski fractals.

The results connect geometric and functional analytic properties of functions.

Abstract

We characterize functions of finite energy in the plane in terms of their traces on the lines that make up "graph paper" with squares of side length $mn$ for all $n$, and certain $\12-$order Sobolev norms on the graph paper lines. We also obtain analogous results for functions of finite energy on two classical fractals: the Sierpinski gasket and the Sierpinski carpet.