  In this work, we give new sufficient conditions for a Littlewood-Paley-Stein
square function and necessary and sufficient conditions for a
Calder\'on-Zygmund operator to be bounded on Hardy spaces $H^p$ with indices
smaller than $1$. New Carleson measure type conditions are defined for
Littlewood-Paley-Stein operators, and we show that they are sufficient for the
associated square function to be bounded from $H^p$ into $L^p$. New polynomial
growth $BMO$ conditions are also introduced for Calder\'on-Zygmund operators.
These results are applied to prove that Bony paraproducts can be constructed
such that they are bounded on Hardy spaces with exponents ranging all the way
down to zero.
