ON THE ATOMIC AND MOLECULAR DECOMPOSITION OF WEIGHTED HARDY SPACES

The purpose of this article is to give another molecular decomposition for members of weighted Hardy spaces, different from that given by Lee and Lin [J. Funct. Anal. 188 (2002), no. 2, 442-460], and to review some overlooked details. As an application of this decomposition, we obtain the boundedness on H-w(p) (R-n) of every bounded linear operator on some L-p0 (R-n) with 1 < p(0) < +infinity, for all weights w is an element of A(infinity) and all 0 < p <= 1 if 1 < r(w)-1/r(w) p(0), or all 0 < p < r(w)-1/r(w) p(0) if r(w)-1/r(w) p(0) <= 1, where r(w) is the critical index of w for the reverse Holder condition. In particular, the well-known results about boundedness of singular integrals from H-w(p) (R-n) into L-w(p) (R-n) and on H-w(p) (R-n) for all w is an element of A(infinity) and all 0 < p <= 1 are established. We also obtain the H-w(p) (R-n)-H-w(q) (R-n) boundedness of the Riesz potential I-alpha for 0 < p <= 1, 1/q = 1/p - alpha/n, and certain weights w.