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In this note, we  solve the inverse nodal problem and Ambarzumyanproblem for the $p$-Laplacian coupled with periodic or anti-periodicboundary conditions. We also extend some results in a previous paperto $p$-Laplacian with $L^1$ potentials,  and for arbitrary linearseparated boundary conditions. There we prove a generalizedRiemann-Lebesgue Lemma which is of independent interest.

AN INVERSE NODAL PROBLEM AND AMBARZUMYAN PROBLEM FOR THE PERIODIC $p$-LAPLACIAN OPERATOR WITH INTEGRABLE POTENTIALS