Several interpolation theorems on martingale Hardy spaces over weighted measure spaces are given. Our proofs are based on
the atomic decomposition of martingale Hardy spaces over weighted measure spaces. As applications of interpolation theorems,
some inequalities of martingale transform operator are obtained.
Some atomic decomposition theorems are proved in vector-valued weak martingale Hardy spaces wpΣα(X), wpQα(X) and wDα(X). As applications of atomic decompositions, a sufficient condition for sublinear operators defined on some vector-valued
weak martingale Hardy spaces to be bounded is given. In particular, some weak versions of martingale inequalities for the
operators f*, S(p)(f) and σ(p)(f) are obtained.
We prove that the maximal conjugate and Hilbert operators are not bounded from the real Hardy space H1 to L1, where the underlying spaces may be over T or R. We also draw corollaries for the corresponding spaces over T2 and R2.
We consider Hausdorff operators generated by a function ϕ integrable in Lebesgue"s sense on either R or R2, and acting on the real Hardy space H1(R), or the product Hardy space H11(RR), or one of the hybrid Hardy spaces H10(R2) and H01(R2), respectively. We give a necessary and sufficient condition in terms of ϕ that the Hausdorff operator generated by it commutes
with the corresponding Hilbert transform.