# Space contained in the Interpolation of $L^\infty$ and the Wiener Algebra $\mathcal{F}(L^1)$

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 $$s\leq d/r$$?