extension of integral preserving positive operators on $L^p$ into $L^q$

Question

Let $(\Omega,\mu)$ be a finite measure space. Let $T \colon L^\infty(\Omega) \to L^\infty(\Omega)$ be a weak* continuous contractive positive operator such that $\int_\Omega T(f)=\int_\Omega f$ for any positive measurable function $f$.

It seems to me that $T$ induces a bounded operator $T \colon L^p(\Omega) \to L^p(\Omega)$ for any $1 \leq p \leq \infty$.

For what $(p,q) \in [1,\infty]$, do we have an induced bounded operator $T \colon L^p(\Omega) \to L^q(\Omega)$ ?

Answer 1

The operator $T$ is bounded from $L^p$ to $L^q$ if $p\geq q$ using the continuous inclusion $L^p\hookrightarrow L^q$. In general, it is not bounded from $L^p$ to $L^q$ if $p<q$. This can be seen by taking $\Omega=[0,1]$ with the Lebesgue measure and $T$ the identity.