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Let us define the $L^p$ norm of a function $f:\mathbb{R} \to \mathbb{R}$ as $$ \|f\|_p:= \left(\int \lvert f \rvert^p\right)^{1/p}. $$

Let us label the following statement by $S\left(p\right)$: $$ \|f\|_p \leq \|g\|_p. $$

My Question: Suppose $p>q>0$. Then, does there exist any interrelationship between $S\left(p\right)$ and $S\left(q\right)$? To be precise, does one implies the other?

I think neither implies the other, yet I cannot find a concrete counterexample (I was working with $p=2$ and $q=1$). Also, I was wondering if such implication holds if we assume stronger assumptions on $f$ and $g$. Any help will be much appreciated. Thank you.

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  • $\begingroup$ Is this supposed to be universal over $f,g$ or for specific $f,g$? In the universal case there is no such implication when the measure space is $\mathbb{R}$ with the Lebesgue measure. $\endgroup$
    – Ian
    Commented May 2, 2019 at 8:38
  • $\begingroup$ @Ian general $f$ and $g$. I think nothing can be said in general setup (though I'm not being able to produce counterexample). As I mentioned in the last paragraph, maybe an implication holds if one restricts $f$ and $g$ to a restricted class of functions (maybe nonnegative $C^1$ functions). But I don't know a proof. $\endgroup$
    – Eugenia
    Commented May 2, 2019 at 8:44
  • $\begingroup$ Try thinking about examples like $f = \chi_{[0,1]}$ and $g = 2^{-\frac{1}{p}} \chi_{[0,2]}$ if your measure space is $\mathbb{R}$ with the Lebesgue measure. This will also tell you that your restriction doesn't make things better by approximating such indicator functions by smooth functions. $\endgroup$
    – Rhys Steele
    Commented May 2, 2019 at 8:51
  • $\begingroup$ There is no interesting relationship on any broad class of functions simply because $C^\infty$ (or even $C^\omega$) is dense in $L_p$ for $p<\infty$. Inequalities both ways can be made to hold on arbitrarily nice functions, even polynomials, just approximate given functions by them in $L_p\cap L_q$. $\endgroup$
    – Conifold
    Commented May 2, 2019 at 8:51

1 Answer 1

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Let $f=I_{[0,1]}$, i.e., the indicator function on $[0,1]$. Similarly, let $g=\frac{1}{\lambda}I_{[0,\lambda]}$. Then $f$ and $g$ have the same $L^1$ norm, but their $L^2$ norms are $1$ and $\sqrt{\frac{\lambda}{\lambda^2}}$. Choosing values for $\lambda$ that are greater than or less than $1$ gives norms both greater than or less than $1$ for $g$.

Therefore, you need to severely restrict your function $f$ and $g$ to have any hope of a relationship between their norms. There are some things you can say if, say, you knew that $f$ had a smaller $L^p$ and $L^q$ norm than $g$ and wanted an inequality on $L^r$ where $r$ is between $p$ and $q$.

Additional information can be found on the Wikipedia page on $L^p$ spaces.

In addition, Hölder's inequality can be used to prove log-convexity of the $L^p$ norm, which is related, but not equivalent, to your question.

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