Let $f\in H^{1/2}(S)$, $g\in H^{3/2}(S)$, $\alpha\in L^\infty(S)$, where $S$ is a closed smooth $n$-dimensional surface (e.g, $n$-dimensional sphere).
Problem: to find such Banach space $W$ that the estimate \begin{gather}\tag{1}\label{1} (\alpha f,g)_{L^2(S)}\leq C\|\alpha\|_{W}\|f\|_{H^{1/2}(S)}\|g\|_{H^{3/2}(S)}, \end{gather} holds; here $C>0$ in independent of $f,g,\alpha$. Evidently, the above estimate holds for $W=L^\infty(S)$, namely, one has $$ (\alpha f,g)_{L^2(S)}\leq \|\alpha\|_{L^\infty(S)}\|f\|_{L^2(S)}\|g\|_{L^2(S)}\leq \|\alpha\|_{L^\infty(S)}\|f\|_{H^{1/2}(S)}\|g\|_{H^{3/2}(S)}.$$ Question: what is the largest space $W$ for which \eqref{1} holds? Or, at least, can one give an example of $W$ satisfying \eqref{1} and being indeed larger than $L^\infty(S)$ (e.g., some Sobolev space with negative index)?