I would like to ask about any characterization of the dual of the intersection $ L^p( \mathbb{R}) \cap L^q( \mathbb{R}^N),\ N \geq 1, $ equipped with the norm sum, that is $$ \vert u \vert_{L^p( \mathbb{R}^N)} + \vert u \vert_{L^q( \mathbb{R}^N)}, $$ where $ 1 \leq p < + \infty,\ 1 \leq q < + \infty. $ Is there any reference about this issue?
Thank you in advance