Let μ be a Radon measure on Rd, which may be non doubling. The only
condition that μ must satisfy is μ(B(x,r))≤Crn, for all x,r and
for some fixed 0<n≤d. Recently we introduced spaces of type BMO(μ)
and H1(μ) which proved to be useful to study the Lp(μ) boundedness of
Calder\'on-Zygmund operators without assuming doubling conditions. In this
paper a characterization of the new atomic space H1(μ) in terms of a grand
maximal operator MΦ​ is given. It is shown that f belongs to H1(μ)
iff f∈L1(μ), ∫fdμ=0 and MΦ​(f)∈L1(μ), as in the
usual doubling situation. The lack of any regularity condition on μ, apart
from the size condition stated above, is one of the main difficulties that
appears when one tries to extend the classical arguments to the present
situation.Comment: 47 page