0
$\begingroup$

This question looks simple at the first glance but ... I have tried to combine the theorems and definitions on $L^p$ spaces to solve this question but I have not been able to do so. I need help to show that, there is a measurable function $$g\in L^ { p_{0}}\setminus \Bigg( \bigcup_{p\neq {p_{0}}}L^p \Bigg)$$ for every $ 0

  • 4
    Try a function of the form $x^a\log^b(x)$ on $(0,1)$ and another function of that form on $(1,\infty)$2011-11-21
  • 5
    The second half of [this faq-entry](http://math.stackexchange.com/faq#howtoask) explains how you can vote on answers and accept them. Also, a more descriptive title would be very nice. How about "A function belonging to only one $L^p$-space"? [Here's a question](http://math.stackexchange.com/questions/55170/) dealing with the case $1 \lt p \lt \infty$.2011-11-21
  • 0
    @ Arturo and t.b, well noted. I new on this site and trying to learn the rules and procedures. I will do my best to do the right thing. Thanks2011-11-21
  • 0
    No you don't...2011-12-11
  • 1
    I'm voting to close as duplicate of the question t.b. links to. Technically it asks for slightly more, but the general case is an immediate corollary of the case for a single $p\in(0,\infty)$ (just take the $p/p_0$ power).2011-12-17

1 Answers 1

1

The answer for $p\geqslant 1$ is given in this theread. Given a $p>0$, take a function $f$ which is only in in $L^{Np}$ where $N$ is such that $Np\geqslant 1$. Then consider $g:=|f|^{1/N}\in L^p$ but not in $L^q$ if $q\neq p$.