A non-homogeneous Hardy-like inequality has recently been found to be closely related to the knowledge of the lowest eigenvalue of a large class of Dirac operators in the gap of their continuous spectrum.
Hardy spaces consisting of adapted function sequences and generated by the q-variation and by the conditional q-variation are considered. Their dual spaces are characterized and an inequality due to Stein and Lepingle is extended.
We prove the continuity of an oscillatory singular integral operator T with polynomial phase P(x,y) on an atomic space related to the phase P. Moreover, we show that the cancellation condition to be imposed on T holds under more general conditions. To that purpose, we obtain a van der Corput type lemma with integrability at infinity.
As a natural extension of Sobolev spaces, we consider Hardy-Sobolev spaces and establish an atomic decomposition theorem, analogous to the atomic decomposition characterization of Hardy spaces. As an application, we deduce several embedding results for Hardy-Sobolev spaces.