Smooth and Nonsmooth Calderón-Zygmund Type Decompositions for Morrey Spaces

It is shown that it is possible to construct an analogue of the Calderón-Zygmund decomposition for the Morrey spaces Mor_λ for the entire interval λ ∈ (0,1]. Moreover, for λ ∈ (1-,1] it is possible to construct a smooth analogue of the Calderón-Zygmund decomposition. The reason why we do not have any smooth analogues for the entire interval λ ∈ (0,1] is related to the following interesting property of cubes in the Whitney decomposition lemma: The sum of the volumes of Whitney cubes to the power λ is equal to infinity for λ ∈ (0,1-(1/n)].