On the generalized Morrey spaces

We show the equivalence of some real variable characterization and wavelet characterization for the generalized Morrey spaces, using a $2n$-dimensional wavelet to analyze an element in $Q_p^{\alpha,q}$ on $\mathbb{R}^n$. We also construct the preduals $P_p^{\alpha,q}$ of $Q_p^{\alpha,q}$ which are generated by some atoms defined by wavelets.