We have the complex interpolation $[H^1,L^p]_\theta=L^q$ for $1<p<\infty$ where $H^1$ is the Hardy space and $\frac{1}{q}=1-\theta+\frac{\theta}{p}$. This raises the obvious question as to whether $[H^1,L^1]_\theta$ is any familiar space.
I'm inclined to say no since obviously $H^1\subset[H^1,L^1]_\theta\subset L^1$ but intuitively there is "little room" for anything interesting to happen between the two spaces.