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!


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.