Applying the complex interpolation method to $L^p$ spaces

I am currently reading about the abstract complex interpolation method, which is a generalization of the Riesz-Thorin theorem. I have been trying to see how to apply it to $L^p$ spaces.

My question is: If you apply the abstract complex interpolation method with exponent $\theta$ to the couple $(L^{p_0},L^{p_1})$, do you get the space $L^p$ ($1/p =(1-\theta)/p_0 + \theta/p_1$)

To keep the question self-contained, I will describe the essential problem I am facing: Let $S$ be the strip $\{z \in \mathbb{C}, 0\leq Re(z)\leq 1\}$. Then, the class of functions $\mathcal{F}(p_0,p_1)$ consists of all functions $f:S\rightarrow L^{p_0}+L^{p_1}$ with the property that

1. $f$ is continuous and bounded on $S$, and weakly analytic in the interior.

2. $f(j+it) \in L^{p_j}, j=0,1$.

Let $$\frac{1}{p(\theta)}=\frac{1-\theta}{p_0} + \frac{\theta}{p_1}$$

For $0\leq \theta \leq 1$, does it follow that $f(\theta) \in L^{p(\theta)}$?

See page $91$ in this article by Bill Johnson.