Sobolev's inequality in central Herz-Morrey-Musielak-Orlicz spaces over metric measure spaces

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.

Keywords:

• Maximal functions
• Riesz potentials
• central Herz-Morrey-Musielak-Orlicz spaces
• non-doubling measure
• metric measure space
• double phase functionals

AMS Subject Classifications:

• Primary 46E35
• Secondary 46E30

Acknowledgments

We would like to express our deep thanks to the referee, who generously provides us an example in Appendix, for his/her carefully reading and many helpful and useful comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).