1. ## Use Hölder's inequality

Let $\displaystyle (X, \mathcal{B}, \mu)$ be a measure space. Use Hölder's inequality to prove that if $\displaystyle \frac{1}{p}+\frac{1}{q} = \frac{1}{r}$ and $\displaystyle r \geq 1$ then for every $\displaystyle f \in L^p(X, \mathcal{B}, \mu)$, $\displaystyle g \in L^q(X, \mathcal{B}, \mu)$, and $\displaystyle fg \in L^r(X, \mathcal{B}, \mu)$ with $\displaystyle || fg ||_r \leq || f ||_p \cdot || g ||_q$.

Let $\displaystyle (X, \mathcal{B}, \mu)$ be a measure space. Use Hölder's inequality to prove that if $\displaystyle \frac{1}{p}+\frac{1}{q} = \frac{1}{r}$ and $\displaystyle r \geq 1$ then for every $\displaystyle f \in L^p(X, \mathcal{B}, \mu)$, $\displaystyle g \in L^q(X, \mathcal{B}, \mu)$, and $\displaystyle fg \in L^r(X, \mathcal{B}, \mu)$ with $\displaystyle || fg ||_r \leq || f ||_p \cdot || g ||_q$.
Simply apply Hölder's inequality with the correct p and q... Write down what you want to prove ($\displaystyle (\int (f g)^r)^{1/r}\leq \cdots$), rewrite it like Holder's inequality ($\displaystyle \int (f^r) (g^r) \leq \cdots$) in order to guess the exponents, and its shouldn't be a problem.