Interpolation space between $L^1\cap L^2$ and $L^1$

Question

In the paper 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!

Answer 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, 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 $\|T f\|_p$.

Answer 2

In the case of a set of finite measure (Bourgain in the quoted paper deals with the case of finite measure (torus)) we have that $\Vert f\Vert_1\leq C\Vert f\Vert_2$ so we actually have $(\infty,1)$ and $(2,2)$ estimate and therefore we have $\Vert Tf\Vert_{p'}\leq C\Vert f\Vert_p$, $1<p<2$, by Marcinkiewicz or Riesz-Thorin.

Edit: This is an answer to an earlier version of the question which was not so detailed and only after a discussion with the author of the question details regarding the dependence of the constant were added so you should take it into account before you decide to downvote my answer.