Given a function or distribution space \(X\) on \(\mathbb R^n\) that admits a Littlewood-Paley decomposition, one can define the homogeneous Besov space \(\dot{B}^{r,s}(X)\) with norm \[ \| f\| _{B^{r,s}(X)} =\left\{\sum_{\nu=-\infty}^\infty \left(2^{s\nu}\| Q_\nu f\| _X\right)^r\right\}^{1/r} \] in which the \(Q_\nu\), corresponding to Fourier multipliers localized on dyadic annuli of radius \(2^\nu\), define a Littlewood-Paley decomposition of the identity: \(I= \sum_{\nu=-\infty}^\infty Q_\nu\). Here, the author takes \(X\) to be the Morrey space \({M}^p_q\) consisting of those locally integrable \(f\) such that for all \(x_0\in \mathbb R^n\) and all \(R>0\), \(R^{n(q/p-1)}\int_{B(x_0;R)}| f| ^q\leq C^q\). The smallest \(C\) for which the inequality holds is the Morrey norm \(\| f\| _{{{M}^p_q}}\). With a special choice of Littlewood-Paley or wavelet decomposition coming from the so-called discrete Calderón reproducing formula, the author denotes the space \(B^{r,s}({M}^p_q)\) by \(\mathcal{N}_{s,p,r}^q\). The main result offers a characterization of the norm on \(\mathcal{N}_{s,p,r}^q\) in terms of wavelet coefficients, namely, if \(f=\sum_Q s_Q \sigma_Q\) in which \(\sigma_Q\) is a suitable \(L^1\)-normalized wavelet localized on the dyadic cube \(Q\), then one has the norm equivalence of \(\| f\| _\ast\) and \(\| f\| _{\mathcal{N}_{s,p,r}^q}\) where \[ \| f\| _\ast^r = \sum_{\nu=-\infty}^\infty \bigg(\sup_{\ell(J)\geq 2^{-\nu}} | J| ^{q/p-1}\sum_{Q\in J_\nu}| Q| ^{1-q/p}| s_Q| ^q\bigg)^{r/q}. \] The cubes \(J,Q\) in the sum are dyadic and \(J_\nu\) denotes the dyadic cubes contained in \(J\) having sidelength \(2^{-\nu}\). The author extends this characterization to the case in which the wavelets \(\sigma_Q\) are replaced by arbitrary molecules that give rise to a suitable partition of unity.
For the entire collection see [Zbl 1013.00026].