Let $L_p^0$ be the mean zero functions in $L_p(G)$, where, say, $G$ is an infinite compact group endowed with normalized Haar measure. Suppose that $T$ is a bounded linear operator on $L_1$ that maps $L_1^0$ into itself and $L_2$ into $L_2$. Suppose that $\|T\|_{L_1 \to L_1} =1$ and $\|T\|_{L_2^0\to L_2^0} <1$.

Q1: If $1<p<2$, must $\|T\|_{L_p^0\to L_p^0} < 1$?

Q2: What if, in addition, $T$ is given by convolution with respect to a probability measure on $G$?

Several papers claim that Q2 has an affirmative answer in such a way that they seem to imply that Q1 also has an affirmative answer. (The claim is expressed by something like “By the Riesz convexity theorem”, but I do not see that it follows from either the theorem or the standard proof of the theorem.)