Bergh and Löfstrom's book Interpolation spaces - An introduction gives a short proof sketch (Thm. 5.5.3, p. 120) of the fact that the complex interpolation space $(L^{p_0}(w_0), L^{p_1}(w_1))_{[θ]}$ is $L^{p_θ}({w_0}^{(1-θ) p_θ/p_0} {w_1}^{θ p_θ/p_1})$. Here $\frac{1}{p_\theta} = \frac{1-\theta}{p_0} + \frac{\theta}{p_1}$
The proof consists of the claim that the map $\mathcal{F}\left(L^{p_0}(w_0), L^{p_1}(w_1)\right) \to \mathcal{F}(L^{p_0}, L^{p_1}), f \mapsto \tilde{f}$ defined by $$ \tilde{f}(z) = f(z) \cdot {w_0}^{(1-z)/p_0} {w_1}^{z/p_1} \tag{1} $$ is an isometric isomorphism between the spaces. Here $\mathcal{F}(X, Y)$ denotes the space of continuous bounded functions from the strip $0 ≤ \operatorname{Re} z ≤ 1$ into $X + Y$ that are holomorphic in the interior and whose norm along the boundary tend to zero at infinity, from which the interpolation space is constructed.
Deducing the theorem from this claim seems straightforward, but I see two problems with the claim itself:
- First, I don't see why the RHS of (1), for a particular value of $z$, is a function in $L^{p_0} + L^{p_1}$. In the case $p_0 = p_1$ this basically already is the whole claim of the interpolation result. And in general, $g \in L^p(w_0) + L^p(w_1)$ does not imply $g \in L^p({w_0}^{1-θ} {w_1}^θ)$.
- Second, I don't see how you can conclude that this is surjective. In fact, if $w_0$ or $w_1$ are zero somewhere this seems to be false.
Any pointers on this?
The isometry of the tilde map on the other hand is immediate from the definition of the norm on the $\mathcal{F}$-spaces.