Prove that any function \(f \in L^1(I) \cap L^2(I)\) on an interval \(I \subset \mathbb{R}\) must belong to \(L^p(I)\) for all \(p \in [1, 2]\).
Solution
Fix \(p\in[1,2]\) and let \(E_1=\{x\in I:\abs{f(x)}\le1\}\) and \(E_2=\{x\in I:\abs{f(x)}>1\}\).
1. \(\abs f^p\le\abs f\) on \(E_1\) and \(\abs f^p\le\abs f^2\) on \(E_2\).
For \(0\le t\le1\) and \(p\ge1\), \(t^p\le t\); for \(t>1\) and \(p\le2\), \(t^p\le t^2\).
2. Q.E.D.
By step 1, \(\int_I\abs f^p\le\int_{E_1}\abs f+\int_{E_2}\abs f^2\le\norm f_{L^1(I)}+\norm f_{L^2(I)}^2<\infty\), so \(f\in L^p(I)\).