Let $1<p<q\leq\infty$ and $J=[0,T)$ for some $T>0$. The measure space we consider is the usual: $J$ together with the appropriate Borel-Sigma-Algebra and the Lebesgue measure. Can you give me an example of a Lipschitz map from $L^p(J)$ to $L^q(J)$?
It is not hard to find a Lipschitz map from $L^p(J)$ to $L^p(J)$ or from $L^q(J)$ to $L^p(J)$ (since $J$ is finite we have $L^q(J)\subset L^p(J)$ with analogue norm estimate). But I don't see an easy example for my situation. The map may also be only locally Lipschitz.
Finally, the definitions I am working with:
1) Let $(A,\textrm{d}_A)$ and $(B,\textrm{d}_B)$ be metric spaces. $f:A\rightarrow B$ is called Lipschitz continuous if there is a constant $M\geq 0$ such that $\textrm{d}_B(f(a_1),f(a_2))\leq M\textrm{d}_A(a_1,a_2)$ for all $a_1,a_2\in A.$
2) $f:A\rightarrow B$ is called locally Lipschitz continuous if for every $a\in A$ there is a neighbourhood $U$ of $a$ such that $f\vert_U$ is Lipschitz continuous.
Thanks for your answers.
The map $$ \phi:L^p(J)\to L^q(J)\\ f \mapsto 0 $$ is globally Lipschitz continuous.