8-2

Question 8-2: Prove the converse of Hölder's inequality for p=1 and \infty. Show also that for 1\leq p\leq \infty, a real-valued measurable f belongs to L^p(E) if fg\in L^1(E) for every g\in L^{p'}(E), 1/p+1/p'=1. The negation is also of interest: if f\notin L^p(E), then there exists g\in L^{p'}(E) such that fg\notin L^1(E). (To verify the negation, construct g of the form \sum a_kg_k for appropriate a_kand g_k, with g_k satisfying \int_E fg_k\rightarrow +\infty.)
Answer: Part 1 has a solution on Math Stackexchange here: https://math.stackexchange.com/questions/4179726/a-certain-converse-of-h%C3%B6lders-inequality-for-l-infty-space
 
Part 2: Show also that for 1\leq p\leq \infty, a real- valued measurable f belongs to L^p(E) if fg\in L^1(E) for every g\in L^{p'}(E), 1/p+1/p'=1.
Lemma 8.1.1. There exists M>0 such that \lVert fg \rVert_1\leq M, for all g\in L^{p'}(E), \lVert g \rVert_{p'}\leq 1.
proof. Suppose not. Then we have a sequence of L^{p'} functions \{g_k\} with \lVert g_k \rVert_{p'}\leq 1 where \int_E|fg_k|>3^k.
Let g=\sum_{k=1}^\infty 2^{-k}|g_k|. Then \lVert g \rVert_{p'}\leq\sum_{k=1}^\infty 2^{-k}=1 but \int_E|fg|=\sum_{k=1}^\infty2^{-k}\int_E|fg_k|>\sum_{k=1}^\infty(\frac{3}{2})^k=\infty,
so fg\notin L^1(E). This is a contradiction.
So \lVert f \rVert_p=\sup_{\lVert g \rVert_{p'}\leq1}\int_E fg\leq\sup_{\lVert g \rVert_{p'}\leq1}\lVert fg \rVert_1\leq M<\infty.
Thus f\in L^p(E).
Part 3: Show if f\notin L^p(E), then there exists g\in L^{p'}(E) such that fg\notin L^1(E).