A characterization of pointwise multipliers on the Morrey spaces (Q2707058)

For \(a\in \mathbb R^n\) and \(r>0\), let \(B(a,r)=\{x\in\mathbb R^n;|x-a|<r\}\). For a measurable set \(E\subset \mathbb R^n\), one denotes by \(|E|\) the Lebesgue measure of \(E\). For \(0<p\leq \infty \) and \(\varphi: (0, \infty)\to (0, \infty)\), let \(L_{p,\varphi}=\{f\in L^p_{\text{loc}}(\mathbb R^n): \|f\|_{p,\varphi<\infty}\}\), where \(\|f\|_{p,\varphi}=\sup_{B(a,r)}\varphi(r)^{-1} (|B(a,r)|^{-1}\int_{B(a,r)}|f(x)|^p dx)^{1/p}\) \((0<p<\infty)\), \(=\sup_{B(a,r)}\varphi(r)^{-1}\) with limits \(\text{ess sup}_{B(a,r)}|f(x)|\). For \(\varphi(r)=r^{(\lambda -n)/p}\) \((0<p<\infty, 0<\lambda <n)\), \(L_{p,\varphi}(\mathbb R^n)\) is the classical Morrey space. The author's main result is: Let \(0<p_2\leq p_1\leq\infty\). Suppose that \(\varphi_i\) \((i=1,2)\) are almost decreasing and that \(\varphi_i(r)r^{n/p_i}\) \((i=1,2)\) are almost increasing. Then, \(\text{PWM}(L_{p_1,\varphi_1}(\mathbb R^n), L_{p_2,\varphi_2}(\mathbb R^n))= L_{p_3,\varphi_3}(\mathbb R^n)\), if and only if \(\varphi_2^{p_2/p_1}/\varphi_1\) \((\varphi_2/\varphi_1\text{ when }p_1=p_2=\infty, 1/\varphi_1\text{ when }p_2<p_1=\infty)\) is almost increasing, where \(1/p_3=1/p_2-1/p_1\) and \(\varphi_3=\varphi_2/\varphi_1\). Here, \(\text{PWM}(X, Y)\) denotes the set of all pointwise multipliers from the function space \(X\) to the function space \(Y\). The sufficiency was already given by the same author in ``Pointwise multipliers on the Morrey spaces'' [Mem. Osaka Kyoiku Univ., III Natur. Sci. Appl. Sci. 46, 1-11 (1997; MR 99b:46045)].