Is this map surjective?

Question

Let $B^1(\mathbb{R},\mathbb{R})$ be the set of all locally integrable functions $f:\mathbb{R}\to \mathbb{R}$ such that $$\sup_{t\in \mathbb{R}} \int_t^{t+1}|f(x)|dx<\infty.$$ Consider the map $A:B^1(\mathbb{R},\mathbb{R})\to L^\infty(\mathbb{R},L^1([0,1],\mathbb{R}))$ defined by $$(Af)(t)(s)=f(t+s).$$ Is $A$ surjective ?

Answer 1

Certainly not. If $g$ is in the image of $A$, it has the property that $x\mapsto g(x)(1-x)$ is constant, which not all elements of $L^\infty(\mathbb R,L^1([0,1],\mathbb R))$ have