  In the setting of spaces of homogeneous type, we give a direct proof of the
local Tb theorem for singular integral operators. Motivated by questions of S.
Hofmann, we extend it to the case when the integrability conditions are lower
than 2, with an additional weak boundedness type hypothesis, which incorporates
some Hardy type inequalities. The latter can be obtained from some geometric
conditions on the homogeneous space. For example, we prove that the monotone
geodesic property of Tessera suffices.
