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I consider two $L^1$ weighted spaces, with $m_1(x) = e + x, m_0 = \ln(e+x)$. It is known that the Riesz-Thorin interpolation theorem holds for $L^p$-weighted space. I have an operator $T: L^1(m_1) \to L^1(m_1)$ bounded with norm $M_1$ and bounded as an operator for $L^1$ to $L^1$, with norm $M$. From the Riesz-Thorin adaptation to this case, I know that for any $\theta \in (0,1)$, the norm of $T$ as an operator from $L^1 (m_1^{\theta})$ to itself is bounded by $M^{1-\theta} M_1^{\theta}$.

What I want to prove is the following: since $1 \leq m_0(x) \leq m_1(x)$ for all $x$, it seems that a result of the same kind of result should exist to prove that $T$ is bounded as an operator from $L^1(m_0)$ to itself. I do believe that $L^1(m_0)$ is an intermediate space for the pair $(L^1, L^1_{m_1})$.

So my two questions are:

1) Is $L^1(m_0)$ indeed an intermediate space for the pair $(L^1, L^1(m_1))$?

2) If so, even if I can not write $m_0$ as $m_1^{\theta} 1^{1-\theta}$ for any value $\theta \in (0,1)$, does the result of Riesz-Thorin still holds ?

I guess on a more basic groud my question is just "Is every intermediate space an interpolation space ? And if not so what would be a counterexample ? "

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