Spaceability of sets of nowhere $L^q$ functions

TL;DR

This paper proves that the set of functions that are nowhere $L^q$ on [0,1], for all q > p, contains a large, structured linear subspace, extending previous results on spaceability of certain function sets.

Contribution

It demonstrates that the set of nowhere $L^q$ functions is spaceable and contains a complemented isometric copy of p, generalizing prior work on spaceability of functions outside certain $L^q$ spaces.

Findings

The set $S_p$ is spaceable and contains a complemented isometric copy of p.

This extends previous results on the spaceability of functions outside $L^q$ spaces.

The result generalizes earlier work on nowhere integrable functions.

Abstract

We say that a function $f:[0,1]\rightarrow \R$ is \emph{nowhere $L^q$} if, for each nonvoid open subset $U$ of $[0,1]$, the restriction $f|_U$ is not in $L^q(U)$. For a fixed $1 \leq p <\infty$, we will show that the set $$ S_p\doteq {f \in L^p[0,1]: f is nowhere $L^q$, for each p<q \leq \infty}, $$ united with ${0}$, contains an isometric and complemented copy of $\ell_p$. In particular, this improves a result from G. Botelho, V. F\'avaro, D. Pellegrino, and J. B. Seoane-Sep\'ulveda, $L_p[0,1]\setminus \cup_{q>p} L_q[0,1]$ is spaceable for every $p>0$, preprint, 2011., since $S_p$ turns out to be spaceable. In addition, our result is a generalization of one of the main results from S. G{\l}\c{a}b, P. L. Kaufmann, and L. Pellegrini, Spaceability and algebrability of sets of nowhere integrable functions, preprint, 2011.