Is there a constant $c>0$ for a sufficiently good function $f : \mathbb R^n \to \mathbb R$, $$\| f^2 \|_2 \leqslant c\| f \|_2 ? $$ If so, I'm wondering the sufficient condition of $f$. What about $\| f^p \|_2 \leqslant c \| f \|_2$ cases($p>1)$?
When does $\| f^2 \|_2 \leqslant c \| f \|_2 $ hold?
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real-analysis
functional-analysis
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4If we replace $f$ by $af$, $a>0$, we see that if $f$ "is good", $af$ cannot "be good" for all $a$. – 2012-07-22
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0@DavideGiraudo : I am not getting what u mean by "good"? do u mean to say that the norm may not be finite or something ? – 2012-07-22
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0@theorem In fact I don't understand what is meant by good in the OP. The class of functions $C(c)$ for which such a property can be true has to be contained in $L^4$ and for a $f$, the set $\{a,af\in C(c)\}$ is bounded. – 2012-07-22
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7The point is that the inequality does not scale properly. For any $f \in L^4$, $\|f^2\|_2$ is finite and $c = \|f^2\|_2/\|f\|_2$ works, but there is no $c$ that works for all multiples of any given nonzero $f$. – 2012-07-22
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12Bamily: Why on earth did you accept this answer? – 2012-07-26