  We give various equivalent formulations to the (partially) open problem about
$L^p$-boundedness of Bergman projections in tubes over cones. Namely, we show
that such boundedness is equivalent to the duality identity between Bergman
spaces, $A^{p'}=(A^p)^*$, and also to a Hardy type inequality related to the
wave operator. We introduce analytic Besov spaces in tubes over cones, for
which such Hardy inequalities play an important role. For $p\geq 2$ we identify
as a Besov space the range of the Bergman projection acting on $L^p$, and also
the dual of $A^{p'}$. For the Bloch space $\SB^\infty$ we give in addition new
necessary conditions on the number of derivatives required in its definition.
