Space of $L^p$ functions not in $L^q$ for all $q\neq p$

Question

Fix $p\geq 1$, and consider the set $S$ of all $f\in L^p(\mathbb R)$ with the property that $f\not\in L^q(\mathbb R)$ whenever $1\leq q\leq\infty$ and $q\neq p$. For example, the function $f$ defined by $$f(x)=\begin{cases} (x\log^2x)^{-1/p}, & \mbox{if } 0<x<1/2\:\text{ or }\:2\leq x; \\ 0, & \mbox{otherwise} \end{cases}$$ is in $S$. I am curious about the size of the set $S$ as a subset of $L^p(\mathbb R)$. In particular, can we determine whether $S$ is of first or second category in the complete metric space $L^p(\mathbb R)$?

Answer 1

Let $T=L^p(\mathbb{R})\setminus S$. I claim that $T$ is of first category, and therefore $S$ is of second category.

By interpolation of Lebesgue spaces, if $f\in L^p\cap L^q$, then $f\in L^r$ for all $r$ between $p$ and $q$. Therefore, $T$ is the countable union of sets $L^p\cap L^q$ where $q$ runs over all rational numbers in $[1, \infty)\setminus \{p\}$.

For each $q\ne p$, the set $L^p\cap L^q$ is the countable union of sets $$ A_{p, q, k} = \{f : \|f\|_p\le k\ \text{ and }\ \|f\|_q\le k\} $$ Thus, it suffices to show $A_{p, q, k}$ is closed in $L^p$ and has empty interior.

Closed: If $f_n\in A_{p, q, k}$ and $f_n\to f$ in $L^p$, then a subsequence converges to $f$ a.e., which by Fatou's lemma implies $\|f\|_q\le \liminf_n \|f_n\|_q \le k$.

Empty interior: pick some $g\in S$. For any $f\in A_{p, q, k}$ and any $\epsilon>0$ we have $f+\epsilon g\notin L^q$, hence $f+\epsilon g\notin A_{p, q, k}$. This shows $f$ is not an interior point of $A_{p, q, k}$.