The local versions of $H^p(ℝ^n)$ spaces at the origin

Let 0 < p ≤ 1 < q < ∞ and α = n(1/p - 1/q). We introduce some new Hardy spaces $HK̇_q^{α,p}(ℝ^n)$ which are the local versions of $H^p(ℝ^n)$ spaces at the origin. Characterizations of these spaces in terms of atomic and molecular decompositions are established, together with their φ-transform characterizations in M. Frazier and B. Jawerth's sense. We also prove an interpolation theorem for operators on $HK̇_q^{α,p}(ℝ^n)$ and discuss the $HK̇_q^{α,p}(ℝ^n)$-boundedness of Calderón-Zygmund operators. Similar results can also be obtained for the non-homogeneous Hardy spaces $HK_q^{α,p}(ℝ^n)$.