Sobolev subspaces of nowhere bounded functions

TL;DR

This paper demonstrates the existence of infinite dimensional subspaces within certain Sobolev spaces where all non-zero functions are nowhere bounded or nowhere in specific Lebesgue spaces, highlighting pathological function behaviors.

Contribution

It establishes the presence of large linear subspaces of nowhere bounded and nowhere $L^q$ functions in subcritical Sobolev spaces, revealing new structural properties.

Findings

Existence of infinite dimensional subspaces of nowhere bounded functions

Existence of subspaces of nowhere $L^q$ functions for certain q

Results hold in subcritical Sobolev spaces

Abstract

We prove that in any Sobolev space which is subcritical with respect to the Sobolev Embedding Theorem there exists a closed infinite dimensional linear subspace whose non zero elements are nowhere bounded functions. We also prove the existence of a closed infinite dimensional linear subspace whose non zero elements are nowhere $L^q$ functions for suitable values of $q$ larger than the Sobolev exponent.