For a given $f in L^{p}((0,1))$, the map $g mapsto B(f,g)$ is a bounded linear functional on $L^{q}((0,1))$. Therefore, (assuming $q in (1,infty)$), there is a $T(f) in L^{q’}((0,1))$ such that

begin{equation*}

B(f,g) = int_{0}^{1} (Tf)(x) g(x) , dx quad text{if} , , g in L^{q}((0,1)).

end{equation*}

Here $q’$ is the dual exponent of $q$ (so that $frac{1}{q} + frac{1}{q’} = 1$), and I have used the well-known result concerning duality in $L^{p}$ spaces (sometimes called Riesz Representation). $T(f)$ is uniquely defined and an exercise shows that the assignment $f mapsto T(f)$ is linear. The boundedness of $B$ implies $T$ is also bounded. (In fact, $|T(f)|_{q’} leq C |f|_{p}$, with $C$ the constant in your question.)

I will have to think about the $q = infty$ case.