On the equality case of the Hölder and Minkowski inequalities

I'm following the book Measure and Integral of Richard L. Wheeden and Antoni Zygmund. This is problem 4 of chapter 8.

Consider $E\subseteq \mathbb{R}^n$ a measurable set. In the following all the integrals are taken over $E$, $1/p + 1/q=1$, with $1\lt p\lt \infty$.

I'm trying to prove that $$\int \vert fg\vert =\Vert f \Vert_p\Vert g \Vert_q$$ if and only if $\vert f \vert^p$ is multiple of $\vert g \vert^q$ almost everywhere.

To do this, I want to consider the following cases: if $\Vert f \Vert_p=0$ or $\Vert g \Vert_q=0$, we are done. Then suppose that $\Vert f \Vert_p\ne 0$ and $\Vert g \Vert_q\ne 0$. If $\Vert f \Vert_p=\infty$ or $\Vert g \Vert_q=\infty$, we are done (I hope). If $0\lt\Vert f \Vert_p\lt\infty$ and $0\lt\Vert g \Vert_q\lt\infty$, proceed as follows.

When we are proving the Hölder's inequality, we use that for $a,b\geq 0$ $$ab\leq \frac{a^p}{p}+\frac{b^q}{q},$$ where the equality holds if and only if $b=a^{p/q}$. Explicitly $$\int\vert fg \vert\leq \Vert f \Vert_p \Vert g \Vert_q \int\left( \frac{\vert f \vert^p}{p\Vert f \Vert_p^p} + \frac{\vert g \vert^q}{q\Vert g \Vert_q^q}\right)=\Vert f \Vert_p \Vert g \Vert_q.$$ From here, we see that the equality in Hölder's inequalty holds iff $$\frac{\vert fg \vert}{\Vert f \Vert_p \Vert g \Vert_q}=\frac{\vert f \vert^p}{p\Vert f \Vert_p^p} + \frac{\vert g \vert^q}{q\Vert g \Vert_q^q}, \text{ a.e.}$$ iff $$\frac{\vert g \vert}{\Vert g \Vert_q}=\left( \frac{\vert f \vert}{\Vert f \Vert_p} \right)^{p/q},\text{ a.e.}$$ iff $$\vert g \vert^q\cdot \Vert f \Vert_p^p=\vert f \vert^p \cdot \Vert g \Vert_q^q,\text{ a.e.}$$ Q.E.D. But, assuming that $\Vert f \Vert_p\ne 0$ and $\Vert g \Vert_q\ne 0$, what about when $\Vert f \Vert_p=\infty$ or $\Vert g \Vert_q=\infty$? How can I deal with it?

In the case of Minkowski inequality, suppose that the equality holds and that $g\not \equiv 0$ (and then $\left( \int \vert f+g \vert^p\right)\ne 0$). I need to prove that $\Vert f \Vert_p$ is multiple of $\Vert g \Vert_q$ almost everywhere. I can reduce to the "Hölder's equality case". I can get $$\vert f \vert^p=\left( \int \vert f+g \vert^p\right)^{-1}\Vert f \Vert_p^p\vert f+g \vert^p$$ $$\vert g \vert^p=\left( \int \vert f+g \vert^p\right)^{-1}\Vert g \Vert_p^p\vert f+g \vert^p$$ almost everywhere, but again, using the finiteness and nonzeroness of $\Vert f \Vert_p$ and $\Vert g \Vert_p$.

Solution 1:

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.

Solution 2:

I'll add some details on the Minkowski inequality (this question is the canonical Math.SE reference for the equality cases, but almost all of it concerns Hölder's inequality).

The standard proof of the Minkowski inequality begins with $$ \begin{align*} \int |f+g|^p &\le \int |f||f+g|^{p-1} + \int |g||f+g|^{p-1} \\ &\le \|f\|_p \| |f+g|^{p-1}\|_q + \|g\|_p \| |f+g|^{p-1}\|_q \end{align*} $$ where $q$ is the conjugate exponent to $p$. This simplifies to $\|f+g\|_p^p \le (\|f\|_p+\|g\|_p) \|f+g\|_p^{p-1} $ as wanted. So, if equality holds, it also holds in the two instances of Hölder's inequality above. Hence $|g|^p$ and $|f|^p$ are both constant multiples of $(|f+g|^{p-1})^q$, which makes them collinear vectors in $L^1$.

Additionally, the equality case requires $|f+g| = |f|+|g|$, which means the signs (or arguments, in the complex case) of $f$ and $g$ must agree a.e. where the functions are not zero. Conclusion: $f$ and $g$ are collinear vectors in $L^p$.