Sobolev inequalities in non-homogeneous central Musielak–Orlicz–Morrey spaces and double phase functionals

In this paper, we give Sobolev inequalities for Riesz potentials of functions in non-homogeneous central Musielak–Orlicz–Morrey spaces [Formula: see text], which is an improvement of our results [T. Ohno and T. Shimomura, Boundedness of maximal operators and Sobolev’s inequality on non-homogeneous central Musielak–Orlicz–Morrey spaces, Mediterr. J. Math. 13 (2016) 3341–3357]. As an application, we study Sobolev inequalities for double phase functionals [Formula: see text].