On $L^1$-Functions with a very Singular Behavior

TL;DR

This paper constructs examples of $L^1$-functions that are unbounded on every open subset, achieved as limits of weighted sums of functions with increasing singular points, revealing complex unbounded behaviors.

Contribution

It introduces a method to generate $L^1$-functions with pervasive unboundedness using limits of functions with dense singularities, and analyzes their properties.

Findings

Functions are unbounded on all open subsets.

Constructed functions have uniform integral bounds.

They lack a pointwise majorant.

Abstract

We give examples of $L^{1}$-functions that are essentially unbounded on every nonempty open subset of their domains of definition. We obtain such functions as limits of weighted sums of functions with the unboundedly increasing number of singular points lying at the nodes of standard compressible periodic grids in $\Bbb R^n$. Moreover, we prove that the latter (basic) functions possess properties of uniform integral boundedness but do not have a pointwise majorant. Some applications of the main results are given.