Product of Lebesgue integrable functions...

**orbit** · May 16th 2011, 06:57 PM

Let f $$\f \in {L^4}\left( R \right),g \in {L^5}\left( R \right)\$ . Check the following:

1) $fg$\fg \in {L^{20/9}}(R)\$$

2) $\[fg \notin {L^6}\left( R \right)\]$

Where
$$\L^P} \buildrel \Delta \over = \left\{ {f:I \to R/f Lebesgue int} \right\}\$

Thanks...

**girdav** · May 17th 2011, 11:38 AM

We have $|f|^{\frac{20}9} \in L^{4\frac 9{20}}$ and $|g|^{\frac{20}9} \in L^{5\frac 9{20}}$ . We use Hölder inequality (with $p =4\frac 9{20}$ and $q =5\frac 9{20}$ ) and we get $|fg|^{\frac{20}9}\in L^{\frac{20}{4\cdot 9}+\frac{20}{5\cdot 9}}=L^1$ .

For the second I don't understand : what about $f(x)=g(x) =e^{-|x|}$ ?

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