Section 3.5 Inequalities
Here we collect a number of inequalities which we will use in later chapters. As usual in this section, $\Omega$ denotes an open subset of $\bR^n$.Subsection 3.5.1 Integral Inequalities
Theorem 3.5.1. Monotone Convergence.
Consider a sequence of measurable functions duch that \(0\leq f_1\leq f_2\leq \dots \) a.e. \(x\in \Omega\text{.}\) Then
\begin{equation*}
\lim_{n\to\infty} \int_\Omega f_n(x) dx = \int_\Omega \left( \lim_{n\to\infty} f_n(x) \right) dx.
\end{equation*}
Theorem 3.5.2. Fatou's Lemma.
Assume \(f_j\geq 0\) are measurable functions, then
\begin{equation*}
\int_\Omega \left( \liminf_{n\to \infty} f_n(x) \right) dx \leq \liminf_{n\to\infty}\int_\Omega f_n(x) dx.
\end{equation*}
Theorem 3.5.3. Dominated Convergence.
Assume \(f_j(x) \to f(x)\) as \(j\to\infty\) for all \(x\in \Omega\text{,}\) and \(|f_j(x)|\leq g(x)\) with \(g\in L^1(\Omega)\text{.}\) Then
\begin{equation*}
\lim_{n\to\infty} \int_\Omega f_n(x) dx = \int_\Omega\left(\lim_{n\to\infty} f_n(x) \right) dx.
\end{equation*}
Proof.
Subsection 3.5.2 Young, Hölder, Minkowski
The Young, Hölder and Minkowski inequalities belong to the standard toolbox of each mathematician who works in analysis. You should never leave your home without these!Theorem 3.5.4. Young.
Consider \(a,b \geq 0, p,q>1\) and \(p\) and \(q\) are conjugate \(\frac{1}{p}+\frac{1}{q} =1.\) Then
\begin{equation*}
ab \leq \frac{a^p}{p} + \frac{b^q}{q}.
\end{equation*}
If \(\ep>0\) we write
\begin{equation*}
ab \leq \ep a^p +\ep^{-\frac{q}{p}} b^q.
\end{equation*}
The most common version of Young's inequality is the version with \(p=q=2\text{:}\)
\begin{equation}
ab \leq \frac{\ep}{2} a^2 + \frac{1}{2\ep} b^2 .\tag{3.5.1}
\end{equation}
Proof.
\begin{equation*}
\frac{a^pb^{-q}}{p} + \frac{1}{q} -a b^{1-q} \geq 0.
\end{equation*}
Since \(p\) and \(q\) are conjugate indices, we have \(1-q = -\frac{q}{p}\) and we write the above inequality as
\begin{equation*}
\frac{\left(a b^{-\frac{q}{p}}\right)^p}{p} + \frac{1}{q} - a b^{-\frac{q}{p}} \geq 0.
\end{equation*}
With the function
\begin{equation*}
f(t) = \frac{t^p}{p} + \frac{1}{q} - t
\end{equation*}
the left hand side is exactly \(f(a b^{\frac{-q}{p}}) \text{.}\) Now it is easy to show that \(f(t)\) has a global minimum of \(0\) at \(f(1)=0\text{.}\) Indeed, \(f'(t) = t^{p-1} -1 \) and \(t=1\) is a critical point. As we have \(f''(1) = p-1 >0\) we find a global minimum at \(1\text{.}\) Finally, to obtain the Peter-Paul estimate (Theorem 3.5.4) we write \(ab = \sqrt{\ep} a \; \frac{b}{\sqrt{\ep}} \) and use Youngs inequality with \(p=q=2\text{.}\)Theorem 3.5.5. Hölder.
Let \((p,q)\) be conjugate indices with \(1<p\leq \infty\) and suppose \(f\in L^p(\Omega)\) and \(g \in L^q(\Omega)\text{.}\) Then \(fg\in L^1(\Omega)\) and
\begin{equation*}
\| fg\|_1 \leq \|f\|_p \|g\|_q .
\end{equation*}
Proof.
\begin{align*}
\int_\Omega \frac{|f|}{\|f\|_p} \frac{|g|}{\|g\|_q} dx \amp\leq \amp \int_\Omega \left[\frac{1}{p} \frac{|f|^p}{\|f\|_p^p} + \frac{1}{q} \frac{|g|^q}{\|g\|_q^q} \right]\; dx\\
\amp=\amp \frac{1}{p} + \frac{1}{q} =1.
\end{align*}
In the case of \(p=\infty\) we simply estimate with the supremum of \(f\text{:}\)
\begin{equation*}
\int|fg| dx \leq \|f\|_\infty \int_\Omega|g| dx = \|f\|_\infty \|g\|_1 .
\end{equation*}
Hence in both cases \(\int|fg| dx\) is bounded and \(fg\in L^1(\Omega)\text{.}\)Theorem 3.5.6. Minkowski.
Assume \(1\leq p <\infty\) and consider \(f,g \in L^p(\Omega)\text{.}\) Then \(f+g \in L^p(\Omega)\) and
\begin{equation*}
\|f+g\|_p \leq \|f\|_p + \|g\|_p.
\end{equation*}
Proof.
\begin{equation*}
|f(x) + g(x) |^p \leq (|f(x)| + |g(x)|)^p \leq c (|f(x) |^p + |g(x)|^p),
\end{equation*}
which implies \(f+g \in L^p(\Omega)\text{.}\) If \((p,q)\) are conjugate, then \((p-1)q=p\) and \(|f+g|^{p-1}\in L^q(\Omega)\text{.}\) Now
\begin{align*}
|f+g|^p \amp=\amp |f+g|^{p-1} |f+g|\\
\amp\leq\amp |f+g|^{p-1}(|f|+|g|)\\
\amp=\amp |f+g|^{p-1}|f| + |f+g|^{p-1}|g|.
\end{align*}
Then, using Hölders inequality on each of these terms, we find
\begin{align*}
\|f+g\|_p^p \amp\leq \amp \| |f+g|^{p-1} \|_q (\|f\|_p+\|g\|_p)\\
\amp=\amp \left(\int_\Omega |f+g|^p dx \right)^{\frac{1}{q}} (\|f\|_p + \|g\|_p)\\
\amp=\amp \|f+g\|_p^{\frac{p}{q}} (\|f\|_p + \|g \|_p),
\end{align*}
leading to
\begin{equation*}
\|f+g\|_p^{p-\frac{p}{q}}\leq \|f\|_p + \|g\|_p.
\end{equation*}
The exponent of the last term on the right hand side is
\begin{equation*}
p- \frac{p}{q} = p \left( 1- \frac{1}{q}\right) = \frac{p}{p} = 1.
\end{equation*}