- Give examples of functions f and g on R so that \(f \in L ^ { 1 } \setminus L ^ { 2 }\) and \(g \in L ^ { 2 } \setminus L ^ { 1 }\)
Solution
Let \(f(x)=x^{-1/2}\mathbf 1_{(0,1]}(x)\) and \(g(x)=x^{-1}\mathbf 1_{[1,\infty)}(x)\).
1. \(\boxed{f=x^{-1/2}\mathbf 1_{(0,1]}}\in L^1(\RR)\setminus L^2(\RR)\).
\(\int_\RR\abs f=\int_0^1x^{-1/2}\,dx=2\), while \(\int_\RR\abs f^2=\int_0^1x^{-1}\,dx=\lim_{\eps\to0^+}(-\log\eps)=\infty\).
2. \(\boxed{g=x^{-1}\mathbf 1_{[1,\infty)}}\in L^2(\RR)\setminus L^1(\RR)\).
\(\int_\RR\abs g^2=\int_1^\infty x^{-2}\,dx=1\), while \(\int_\RR\abs g=\int_1^\infty x^{-1}\,dx=\lim_{R\to\infty}\log R=\infty\).
3. Q.E.D.
Steps 1 and 2 give the two examples.