1

I came up with this when trying to prove embeddings of $L^p(0,1)$ spaces. Suppose $U =(0,1)$ (or more generally a bounded subset in $\Re^n$). Is the mapping $\phi = p \mapsto \|f\|_{L^p(U)}, \phi: [1,q] \rightarrow \Re$ continuous? Any ideas or counter examples?

Edit: This is not a duplicate with Is $p\mapsto \|f\|_p$ continuous? because of different assumption on $f$!

Mundo
  • 127
  • What is restriction on $f$? It belongs to all $L^p$ spaces? Also what is $U$? – BigM Oct 04 '17 at 17:17
  • 1
    The point was to prove embedding $L^q(0,1) \subset L^p(0,1)$ for $1 \leq p \leq q < \infty.$ So U here would be interval (0,1). – Mundo Oct 04 '17 at 18:01
  • It's a fine question but the application seems a bit strange. If you know Hölder's inequality (or just Jensen's) then it is easy to show $|f|_p\leq |f|_q$ for $U=(0,1)$ and $1\leq p\leq q$. How would continuity help? – Dap Oct 04 '17 at 21:20
  • Similar question answered at https://math.stackexchange.com/questions/133773/is-p-mapsto-f-p-continuous - it doesn't cover continuity at the endpoint $q$ as stated, but the argument does work for the endpoint (just dominated convergence theorem) – Dap Oct 04 '17 at 21:27
  • When Hölder implies the finiteness of $|f|_1$ provided that $|f|_q < \infty$, then finiteness of $|f|_p$ would be a direct consequence of continuity, when $p \in [1,q]$. – Mundo Oct 04 '17 at 21:43
  • It is not exactly a duplicate as I don't assume the condition $|f|_p < \infty$ for all $p \in [1, q]$, rather it is my conclusion. – Mundo Oct 04 '17 at 21:50

0 Answers0