Heat capacities (C) of He-3 submonolayer solids adsorbed on a graphite
surface are measured down to 100 mu K, a factor of twenty lower tempe
ratures than previous work. At areal densities near the commensurate r
oot 3 x root 3 solid (6.4 nm(-2)), an anomalous temperature dependence
, C proportional to 1/T, is observed in a wide temperature range over
two orders of magnitude (0.1 less than or equal to T less than or equa
l to 20mK). Similar behavior was observed for the commensurate root 7
x root 7 solid in the second-layer with the same density, suggesting a
common microscopic mechanism. It is, however, a puzzle that the root
3 x root 3 solid is ferromagnetic according to recent magnetization me
asurements by ISSP group, while the root 7 x root 7 solid is known to
be antiferromagnetic. Possible explanations for this conflict, such as
vacancy effects or multiple-spin exchanges modulated by a substrate p
otential corrugation, are discussed.