We present ALMA observations of the 2M1207 system, a young binary made of a brown dwarf with a planetarymass
companion at a projected separation of about 40 au. We detect emission from dust continuum at 0.89 mm and
from the J = 3 - 2 rotational transition of CO from a very compact disk around the young brown dwarf. The
small radius found for this brown dwarf disk may be due to truncation from the tidal interaction with the planetarymass
companion. Under the assumption of optically thin dust emission, we estimate a dust mass of 0.1 M⊕ for the
2M1207A disk and a 3σ upper limit of ∼1 MMoon for dust surrounding 2M1207b, which is the tightest upper limit
obtained so far for the mass of dust particles surrounding a young planetary-mass companion. We discuss the
impact of this and other non-detections of young planetary-mass companions for models of planet formation
that predict circumplanetary material to surround these objects.