The representation of a linear continuous map $T:L^p(\Omega) \to L^q(\Omega)$

Question

Let $(\Omega, \mathcal F, \mu)$ be a $\sigma$-finite measure space. Let $r, q, p \in (1, \infty)$ such that $$ \frac{1}{q} = \frac{1}{r} + \frac{1}{p}. $$

Let $T:L^p(\Omega) \to L^q(\Omega)$ be linear continuous. I would like to ask if there is $a \in L^r(\Omega)$ such that $$ Tu = au \quad \text{a.e.} \quad \forall u \in L^p(\Omega). $$

Thank you so much for your elaboration!

Answer 1

If I understand your question correctly, you want to know if continuous linear maps between $L^p$ and $L^q$ are necessarily given by point-wise multiplication?

If so, I am afraid that the answer is "no". For example, if $T$ is merely a translation, then it cannot be a point-wise product. So, say, with $p = q = 2$ and $r = +\infty$, the operator $T : f \mapsto f(1+\cdot)$ is linear and continuous but for every $a \in L^\infty(\mathbb{R})$ you can find $f \in L^2(\mathbb{R})$ such that $Tf \neq af$.