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Let $\ell^p$ be the space of sequences with power $p$ summable to $\ell^\infty$, $L^p = L^p(\mathbb{R^d})$ be the Lebesgue spaces and $\mathcal{F}$ be the Fourier $d$-dimensional Fourier transform.

My problem is the following, I have a linear operator $T$ continuous from $L^\infty$ to $\ell^2$ and from $\mathcal{F}(L^1)$ to $\ell^\infty$, and I am looking for the sharpest space $X_p$ such that $T : X_p \to \ell^p$ for $p\in(2,\infty)$ (i.e. I would like the space $X_p$ to have the less possible regularity constaints).

What I have been able to get is telling that $B^0_{\infty,1}⊂L^∞$ and $B^{d/2}_{2,1}⊂ \mathcal{F}(L^1)$, so that by complex interpolation $$ B^{d/r}_{r,1} ⊂ X_p \qquad \text{ with } r=\frac{2\,p}{p-2}. $$

Is it possible to get a sharper estimation? For example decrease the $d/r$ exponent, get a bigger second index than $1$ or get a Bessel-Sobolev space $H^{s,r}$ with $s\leq d/r$?

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