HERZ–MORREY SPACES ON THE UNIT BALL WITH VARIABLE EXPONENT APPROACHING AND DOUBLE PHASE FUNCTIONALS - MIZUTA, YOSHIHIRO; OHNO, TAKAO; SHIMOMURA, TETSU (Nagoya mathematical journal, 2021)
Our aim in this paper is to deal with integrability of maximal functions for Herz–Morrey spaces on the unit ball with variable exponent $p_{1}(· )$ approaching $1$ and for double phase functionals $\unicode[STIX]{x1D6F7}_{d}(x,t)=t^{p_{1}(x)}+a(x)t^{p_{2}}$ , where $a(x)^{1/p_{2}}$ is nonnegative, bounded and Hölder continuous of order $\unicode[STIX]{x1D703}\in (0,1]$ and $1/p_{2}=1-\unicode[STIX]{x1D703}/N>0$ . We also establish Sobolev type inequality for Riesz potentials on the unit ball.