In the [paper](http://math.mit.edu/classes/18.158/bourgain-restriction.pdf) of Bourgain, the way equation (3.78) is deduced from (3.69) and (3.76) seems via the following interpolation result. Let $(X,\mu)$ and $(Y,\nu)$ be two measure spaces and let $T$ be a linear operator from $L^1(X,\mu)+L^2(X,\mu)$ to $L^\infty(Y,\nu)+L^2(Y,\nu)$ such that 


\begin{align}
\|Tf\|\_{L^\infty}&\leq A\|f\|\_{L^1}, \\
\|Tf\|\_{L^2}&\leq B\|f\|\_{L^2}+D\|f\|\_{L^1}.
\end{align}


For $0<\theta<1$, let $p=\frac{1-\theta}{1}+\frac{\theta}{2}$, $\frac{1}{p}+\frac{1}{q}=1$. Then for some constant $C$ independent of $A,B,D$, we have 
$$\|Tf\|\_{L^{q}}\leq C(A^{1-\theta}B^\theta\|f\|\_{L^p}+A^{1-\theta}D^\theta\|f\|\_{L^1}).$$


By scaling the measures $\mu$ and $\nu$ as well as the linear operator $T$, one can reduce it to the case when 
\begin{align}
A=B=D=1.
\end{align}


I tried to prove it by looking for the complex interpolation space between $L^1$ and $L^2\cap L^1$, as well as to directly prove it as to prove the Marcinkiewicz's interpolation inequality. But I could not find the correct proof. Any help is appreciated. Thank you!



Interpolation space between $L^1\cap L^2$ and $L^1$
As requested, I post my comment as an answer (although this is not a true answer, just a possibly useful reference; feel free to edit it if this approach works out).


In Section 3 of the article [*Interpolation of sum and intersection spaces of $L^q$-type and applications to the Stokes problem in general unbounded domains*](https://link.springer.com/article/10.1007/s11565-011-0140-6), P.F. Riechwald studies interpolation between spaces $L^2 \cap L^q$ for $q \geqslant 2$ (and also $L^2 + L^q$ for $q \leqslant 2$). His main tool is Theorem 4, which I reproduce below.



> 
> **Theorem.** Let $\Omega \subseteq \mathbb{R}^n$ be a domain and let $f \in L^1(\Omega) + L^\infty(\Omega)$ be a given and fixed function. Then there exist linear maps
>  $$ S\_1 : L^1(\Omega) + L^\infty(\Omega) \to L^1((0, 1)) , \qquad S\_2 : L^1(\Omega) + L^\infty(\Omega) \to \ell^\infty $$
>  and
>  $$ T\_1 : L^1((0, 1)) \to L^1(\Omega) + L^\infty(\Omega) , \qquad T\_2 : \ell^\infty \to L^1(\Omega) + L^\infty(\Omega) $$
>  satisfying the equality
>  $$ f = T\_1 S\_1 f + T\_2 S\_2 f $$
>  almost everywhere. Moreover, these maps satisfy the estimates
>  $$ \|S\_1 u\|\_{L^p((0, 1))} \leqslant \|u\|\_{L^p(\Omega)} , \qquad \|S\_2 u\|\_{\ell^p} \leqslant \|u\|\_{L^p(\Omega)} $$
>  and
>  $$ \|T\_1 u\|\_{L^p(\Omega)} \leqslant \|u\|\_{L^p((0, 1))} , \qquad \|T\_2 u\|\_{L^p(\Omega)} \leqslant \|u\|\_{\ell^p} $$
>  for all $1 \leqslant p \leqslant \infty$ and all $u$ in the respective $L^p$-spaces.
> 
> 
> 


I suppose the argument used in the proof of Theorem 3 implies that the complex interpolation space between two spaces $L^1(\Omega) \cap L^p(\Omega)$ is again a space of this form. This, in turn, should imply the desired bound on $\|T f\|\_p$.


