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

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

2010-10-04

arxiv.org/abs/1010.0570v1

Mathematics

Classical Analysis and ODEs

23 pages

Scientific paper

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.

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