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Loosely speaking, I would like to know whether membership in some Lebesgue space $L^p$ is stable under small perturbations of the exponent $p$.

Let $\Omega \subseteq \mathbb R^n$ be a bounded domain and let $L^p(\Omega)$ denote the Lebesgue space for exponent $1 \leq p < \infty$. Then $L^q(\Omega) \subseteq L^p(\Omega)$ for all $q > p$.

Let us now suppose that $f \in L^p(\Omega)$. Do we have $f \in L^q(\Omega)$ for any $q < p+\epsilon$ for some choice of $\epsilon > 0$?

If not, then I hope to see an example of a function that is "hard" in some $L^p$ space. Most example that I have encountered (e.g., such as powers of the form $\|x\|^{-\gamma}$) leave some wiggle room for the choice of exponent $p$.

If anything more can be said about the possible range of the exponent, even better. Clearly, functions in Sobolev spaces have some flexibility, as per the Sobolev embedding theorem.

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    $\begingroup$ That this is wrong can be seen by Baire's theorem. This is rather a little exercise than research level. $\endgroup$
    – Jochen Wengenroth
    Commented Oct 24 at 15:30
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    $\begingroup$ If a proper vector space $V$ of a Banach space $X$ is itself a Banach space with respect to a stronger norm, then $V$ is meagre in $X$. Hence, a countable union of such spaces is meagre, too. Thus, $\bigcup_{q > p} L^q = \bigcup_{n \in \mathbb{N}} L^{p+\frac{1}{n}}$ is meagre in $L^p$. (Edit: Ok, the other @Jochen was again slightly faster than me. ;-) ) $\endgroup$
    – Jochen Glueck
    Commented Oct 24 at 15:32
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    $\begingroup$ $f(t)=t^{-1}(\log t/2)^{-2}$ on $[0,1]$ belongs yo $L^1$ but not to $L^p$ with $p>1$ $\endgroup$
    – Fedor Petrov
    Commented Oct 24 at 15:40
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    $\begingroup$ Presumably you know that for each $q > p$, it is possible to find a function that is in $L^p$ but not in $L^q$ (as $|x|^{-\gamma}$ or the like). So for each $n$, we may find $f_n \in L^p \setminus L^{p+1/n}$; by replacing $f_n$ with $|f_n|$ we can assume $f_n \ge 0$, and by scaling $f_n$ we can assume $\|f_n\|_{L^p} \le 2^{-n}$. Now set $f = \sum_n f_n$. The sum converges in $L^p$, so we have $f \in L^p$, but $f \ge f_n \notin L^{p+1/n}$ for all $n$, so $f \notin L^q$ for any $q > p$. $\endgroup$
    – Nate Eldredge
    Commented Oct 24 at 16:22
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    $\begingroup$ @AlpinistKitten No need for apologies. You might consider asking on MathStackExchange before posting such a question here on MO. $\endgroup$
    – Jochen Wengenroth
    Commented Oct 24 at 18:18

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