Prove Holder’s inequality: let \(f\in L^p, g\in L^q\) with \(p, q\) conjugate, and show that \[ \pnorm{fg}p \leq \pnorm{f}p \cdot \pnorm{g}q .\]
Prove Minkowski’s Inequality: \[ 1\leq p < \infty \implies \pnorm{f+g}{p} \leq \pnorm{f}{p}+ \pnorm{g}{p} .\] Conclude that if \(f, g\in L^p(\RR^n)\) then so is \(f+g\).
Let \(X = [0, 1] \subset \RR\).
Give a definition of the Banach space \(L^\infty(X)\) of essentially bounded functions of \(X\).
Let \(f\) be non-negative and measurable on \(X\), prove that \[ \int_X f(x)^p \,dx \converges{p\to\infty}\to \begin{dcases} \infty \quad\text{or} \\ m\qty{\theset{\inverseof{f}(1)}} \end{dcases} ,\] and characterize the functions of each type
Solution
Part (c)2. Let \(f \ge 0\) be measurable on \(X = [0,1]\).
1. \(\int_X f^p = \int_{\theset{f < 1}} f^p + m\theset{f = 1} + \int_{\theset{f > 1}} f^p\).
\(X\) is the disjoint union of \(\theset{f < 1}\), \(\theset{f = 1}\) and \(\theset{f > 1}\), and \(f^p = 1\) on \(\theset{f = 1}\).
2. \(\int_{\theset{f < 1}} f^p \to 0\) as \(p \to \infty\).
On \(\theset{f < 1}\), \(f^p \to 0\) pointwise and \(0 \le f^p \le 1\), and \(m(X) = 1\), so dominated convergence applies.
3. \(\int_{\theset{f > 1}} f^p \to \infty\) if \(m\theset{f > 1} > 0\), and it is \(0\) if \(m\theset{f > 1} = 0\).
If \(m\theset{f > 1} > 0\), then \(m\theset{f > 1 + 1/k} > 0\) for some \(k\), and \(\int_{\theset{f > 1}} f^p \ge (1 + 1/k)^p\, m\theset{f > 1 + 1/k} \to \infty\).
4. Q.E.D.
By steps <1>1–<1>3, \(\int_X f^p \to m\theset{f = 1} = m(f^{-1}(1))\) when \(f \le 1\) a.e., and \(\int_X f^p \to \infty\) when \(m\theset{f > 1} > 0\).