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  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.

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  • $\begingroup$ I would guess that there is no counterexample for 1. For 2, you can take $f \in L^{2,\infty}\setminus L^2$, $g \in (L^1)^* \cap (L^2)^*$ and the operator $T x = g(x) \, f$ for all functions $x$. $\endgroup$ – gerw Sep 7 '18 at 7:54

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