ABSTRACT

We give the boundedness of the Hardy-Littlewood maximal operator $M_{\lambda }$, $\lambda \ge 1$, on central Herz-Morrey-Musielak-Orlicz spaces ${\mathcal{H}}^{\mathrm{\Phi },q,\omega }\left(X\right)$ over bounded non-doubling metric measure spaces and to establish a generalization of Sobolev's inequality for Riesz potentials $I_{\alpha ,\tau }f$, $\tau \ge 1$, $\alpha >0$, of functions in such spaces. As an application and example, we obtain the boundedness of $M_{\lambda }$ and $I_{\alpha ,\tau }$ for double phase functionals Φ such that $\mathrm{\Phi }(x,t)=t^{p\left(x\right)}+a\left(x\right)t^{q\left(x\right)},x\in X,t\ge 0$. These results are new even for the doubling metric measure setting.

摘要

我们给出了Hardy-Littlewood极大算子的有界性 ${\mathrm{中号}}_{\lambda }$， $\lambda \ge \mathrm{1个}$，位于Herz-Morrey-Musielak-Orlicz中央空间 ${\mathcal{H}}^{\mathrm{\Phi }，q，\omega }（X）$ 超越有界非加倍度量空间并建立Sobolev对Riesz势不等式的推广 ${\mathrm{一世}}_{\alpha ，\tau }F$， $\tau \ge \mathrm{1个}$， $\alpha >0$在这种空间中的功能。作为一个应用程序和示例，我们获得了${\mathrm{中号}}_{\lambda }$ 和 ${\mathrm{一世}}_{\alpha ，\tau }$ 用于双相功能Φ $\mathrm{\Phi }（X，Ť）={Ť}^{p（X）}+\mathrm{一种}（X）{Ť}^{q（X）}，X\in X，Ť\ge 0$。这些结果是新的，即使对于度量标准度量设置加倍。