# Weak $L^1$ as real interpolation space between $L^p$-spaces?

Let $\Omega$ be a measure space. We denote $L^{p,q}$ the usual Lorentz space. We use a real interpolation method $(\cdot,\cdot)_{\theta,q}$.

Suppose $1\leqslant p,q\leqslant \infty$. I know that if $0<\theta<1$ and $1/p=1-\theta$, we have $$L^{p,q}=(L^1,L^\infty)_{\theta,q}$$ (with equivalent norms).

Do we have a similar formula for $L^{1,\infty}$?

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