Record Detail

[1, Proposition 3.14] states as follows: Let H1, H2 be any Hardy spaces obtained via an
atomic decomposition. Any linear operator mapping the atoms of H1 into a bounded
set in H2 has a bounded extension from H1 into H2.
M. Bownik [2], based on an example of Y. Meyer in Meyer, Taibleson and Weiss [6],
showed that the Hardy space norm and the finite Hardy space norm may not be
equivalent on the finite Hardy space (defined by restricting atomic decompositions to
be finite sums and the atoms are L∞-atoms in the sense of Coifman-Weiss). Hence,
Proposition 3.14 is not correct as stated.
The paper by Meda, Sj¨ogren and Vallarino [4] establishes, among other things,
that if one replaces L∞ atoms by L2 atoms, the equivalence holds. Hence, for an
operator to have a bounded extension it suffices it is bounded on L2-atoms. Moreover,
the extension coincides with the original operator on H1 ∩ L2. So Proposition 3.14 is
correct if H1 is the original Coifman-Weiss Hardy space and atoms are L2-atoms. As
the atoms in [1] are L2-atoms on a space of homogeneous type, this applies directly
to the spaces Hz 1(X) and (with little extra work) Hr 1(X) defined in [1]: the maximal regularity operator and its adjoint have the boundedness property announced in
Theorem 2.1 there.