# Interpolation between log and polynomials using Riesz-Thorin

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 ? "