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Assume that $ g = g(x,y) \in L^1((0,2\pi)\times (0,2\pi)) $ and satifies \begin{equation*} \int^{2\pi}_0 \alpha'(x) g(x,y) dx = A(\alpha), \ \forall \alpha \in C^\infty_0(0,2\pi), \end{equation*} where $A(\alpha) $ is a constant which only depends on function $ \alpha $. I think it could yield that \begin{equation*} g(x,y) = c_0 p(x) +c_1 q(y) + c_2, \ a.e. \ (x,y) \in \ (0,2\pi)^2 \end{equation*} With observation \begin{equation*} \int^{2\pi}_0 \alpha'(x) (q(y) + C)dx = 0. \end{equation*} However, I feel uncertain on the deduction. Could anybody help? This question could be seen as a question for the detail in (https://mathoverflow.net/questions/330529/generalization-of-du-bois-reymond-lemma-into-dimension-2).

Thank you for kindly help in advance!

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