Inserted: 3 oct 2021
Journal: Real Anal. Exchange
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.
Keywords: Sobolev spaces, Nowhere bounded functions, Sobolev Embedding