Sur la conjecture faible de Greenberg dans le cas abélien $p$-décomposé

Let $p$ be an odd prime. For any CM number field $K$ containing a primitive $p^{\rm th}$-root of unity, class field theory and Kummer theory put together yield the well known reflection inequality $\lambda^+ \leq \lambda^-$ between the ``plus'' and``minus'' parts of the $\lambda$-invariant of $K$. Greenberg's classical conjecture predicts the vanishing of $\lambda^+$. We propose a weak form of this conjecture: $\lambda^+ = \lambda^-$ if and only if $\lambda^+ = \lambda^- = 0$, and we proveit when $K^+$ is abelian, $p$ is totally split in $K^+$, and certain (mild) conditions on the cohomology of circular units are satisfied.