In this paper, we establish the decompositions of Hardy-Morrey spaces in terms of atoms concentrated on dyadic cubes, which have the same cancellation properties of the classical Hardy spaces.

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Abstract. We consider the Neumann problem for the Schrödinger equations −∆u + Vu = 0, with singular nonnegative potentials V belonging to the reverse Hölder class B n , in a connected Lipschitz domain Ω ⊂ R n . Given boundary data g in H p or L p for 1 − ǫ < p ≤ 2, where 0 < ǫ < 1 n , it is shown that there is a unique solution, u, that solves the Neumann problem for the given data and such that the nontangential maximal function of ∇u is in L p (∂Ω). Moreover, the uniform estimates are found.

In this paper, the author establishes the decomposition of Morrey type Besov-Triebel spaces in terms of atoms and molecules concentrated on dyadic cubes, which have the same smoothness and cancellation properties as those of the classical Besov-Triebel spaces. The results extend those of M. Frazier, B. Jawerth for BesovTriebel spaces and those of A. L. Mazzucato for Besov-Morrey spaces.