Counterexample in real interpolation

Question

1. Can we find a linear operator $T$ that maps $L^1$ to $L^{2,\infty}$ and $L^2$ to $L^{2,\infty}$ such that $T$ does not map any $L^p$, $1<p<2$ to $L^{2,\infty}$?
2. Can we find a linear operator $T$ that maps $L^1$ to $L^{2,\infty}$ and $L^2$ to $L^{2,\infty}$ such that $T$ does not map any $L^p$, $1<p<2$ to $L^{2}$?

The motivation for my questions is that in the real interpolation theorem, we need to assume $p_0\neq p_1$ and $q_0\neq q_1$; the condition $p_0\neq p_1$ can sometimes be relaxed, but $q_0\neq q_1$ seems always necessary.