Let (X,A,μ) be any measure space and let u∈⋂p∈[1,∞]Lp(μ). Then lim
I have already proved the inequality \|u\|_{\infty} \leq \liminf_{p\to \infty} \| u \|_p, so I still need to prove \limsup_{p\to \infty} \| u \|_p \leq \| u \|_{\infty}.
Following a hint, I have proved that for any sequence q_n \to \infty one has \|u\|_{p+q_n} \leq \|u\|_{\infty}^{q_n/(p+q_n)} \cdot \|u\|_p^{p/(p+q_n)}.
How can we conclude from this that \limsup_{p\to \infty} \| u \|_p \leq \| u \|_{\infty}?
No comments:
Post a Comment