lp-spacesmeasure-theorynormed-spacesreal-analysis
Let $f\in L^p(\mathbb{R}),g\in L^q(\mathbb{R})$ ($1\le p,q<\infty:\frac 1 p+\frac 1 q=1$). Prove that $L^\infty(\mathbb{R}) \ni f\ast g$ (the convolution of them) and also prove that $$\Vert f\Vert_p \Vert g\Vert_q\ge\Vert f\ast g\Vert_\infty$$
I have no idea how to prove the convolution is essentially bounded but the second requirement reminds me Hölder's inequality but $\Vert f\Vert_1\not\geq\Vert f\Vert_\infty$ so also here I'll be glad for a hint.
On leo's request I'm posting my comment as an answer.
Your treatment of the equality cases of Hölder's and Minkowski's inequalities are perfectly fine and clean. There's a small typo when you write that $\int|fg| = \|f\|_p\|g\|_q$ if and only if $|f|^p$ is a constant times of $|g|^q$ almost everywhere (you write the $p$-norm of $f$ and the $q$-norm of $g$ instead).
The case where either one $\|f\|_p$ or $\|g\|_q$ (or both) are infinite isn't part of this exercise and simply wrong. You can trisect $E = F \cup G \cup H$ into disjoint measurable sets of positive measure, take $f$ not $p$-integrable on $F$ and zero on $G$, take $g$ not $q$-integrable on $G$ and zero on $F$ and choose $fg$ non-integrable on $H$. Then certainly no power of $|f|$ is a constant multiple of a power of $|g|$ and vice versa, even though equality holds in the Hölder inequality.
A very nice “blackboard summary” of the equality case (for finite sequences) is given in Steele's excellent book The Cauchy–Schwarz Master Class. Let $a = (a_1,\ldots,a_n) \geq 0$ and $b = (b_1, \ldots, b_n) \geq 0$ and let $\hat{a}_i = \dfrac{a_i}{\|a\|_p}$ and $\hat{b}_i = \dfrac{b_i}{\|b\|_q}$. Then your argument is subsumed by the diagram (with an unfortunate typo in the upper right corner—no $p$th and $q$th roots there):
Mimicking this for functions, let us write $\hat{f} = \dfrac{|f|}{\|f\|_p}$ and $\hat{g} = \dfrac{|g|}{\|g\|_q}$ (assuming of course $\|f\|_p \neq 0 \neq \|g\|_q$), so $\int \hat{f}\vphantom{f}^p = 1$ and $\int \hat{g}^q =1$ and thus your argument becomes $$ \begin{array}{ccc} \int |fg| = \left(\int|f|^p\right)^{1/p} \left(\int|g|^q\right)^{1/q} & & |f|^p = |g|^q \frac{\|f\|_{p}^p}{\|g\|_{q}^q} \text{ a.e.}\\ \Updownarrow\vphantom{\int_{a}^b} & & \Updownarrow \\ \int \hat{f}\,\hat{g} = 1 & & \hat{f}\vphantom{f}^p = \hat{g}^q \text{ a.e.} \\ \Updownarrow\vphantom{\int_{a}^b} & & \Updownarrow \\ \int \hat{f}\,\hat{g} = \frac{1}{p} \int \hat{f}\vphantom{f}^p + \frac{1}{q} \int \hat{g}^q & \qquad \iff \qquad & \hat{f}\,\hat{g} = \frac{1}{p} \hat{f}\vphantom{f}^p + \frac{1}{q} \hat{g}^q \text{ a.e.} \end{array} $$
I suggest that you draw a similar diagram for the equality case of Minkowski's inequality.
Best Answer
You need to use Hölder's inequality. Here is a start
which proves boundedness. It easily follows that $f*g\in L^{\infty}$.