# Thread: Product of Lebesgue integrable functions...

1. ## Product of Lebesgue integrable functions...

Let f$\displaystyle$\f \in {L^4}\left( R \right),g \in {L^5}\left( R \right)\$. Check the following: 1)$\displaystyle fg$\fg \in {L^{20/9}}(R)\$$2) \displaystyle $fg \notin {L^6}\left( R \right)$$$ Where$\displaystyle $\L^P} \buildrel \Delta \over = \left\{ {f:I \to R/f Lebesgue int} \right\}\$

Thanks...

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

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