Multipliers and Morrey spaces
The author gives a necessary and sufficient condition of pointwise multipliers between Morrey spaces. The Morrey space \(\dot{M}^{p,q}=\dot{M}^{p,q}(\mathbb{R}^d)\) is defined by \[ \sup_{Q \in \mathcal{Q}}R_{Q}^{d/q-d/p}\biggl(\int_{Q}|f(x)|^p \;dx \biggr)^{1/p}< \infty \] with the norm \(||f||_{\dot{M}^{p,q}}=\sup_{Q \in \mathcal{Q}}R_{Q}^{d/q-d/p}\biggl(\int_{Q}|f(x)|^p \;dx \biggr)^{1/p} \) where \(\mathcal{Q}\) is the collection of cubes \(Q\) with its size \(R_{Q}\). Let \(\mathcal{M}(\dot{M}^{p_{0},q_{0}}\rightarrow \dot{M}^{p,q})\) be the set of pointwise multipliers from \(\dot{M}^{p_{0},q_{0}}\) to \(\dot{M}^{p,q}\) with the norm \[ ||f||_{\mathcal{M}(\dot{M}^{p_{0},q_{0}}\rightarrow \dot{M}^{p,q})}= \sup_{||g||_{\dot{M}^{p_{0},q_{0}}}\leq 1}||fg||_{\dot{M}^{p,q}}. \] Theorem. Let \(1 < p \leq q\) and \(1 < p_{0} \leq q_{0} < \infty\) with \(1/p=1/p_{0}+1/p_{1}\) and \(1/q=1/q_{0}+1/q_{1}\). Then \(\dot{M}^{p_{1},q_{1}}=\mathcal{M}(\dot{M}^{p_{0},q_{0}}\rightarrow \dot{M}^{p,q})\) if and only if \(q_{1}/p_{1} \geq q_{0}/p_{0}\) with equality of norms. In the case \(q_{1}/p_{1} < q_{0}/p_{0}\), the author constructs a counter-example based on a fractal set of \(\mathbb{R}^d\) with Hausdorff dimension \(\beta =d-dp_{1}/q_{1}\). Furthermore, the author shows that this fractal set also allows us to give simple counter-examples for trace inequalities and interpolation of operators on Morrey spaces.
scientific article; zbMATH DE number 6160848