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

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.