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We prove a generalized Trudinger's inequality for Riesz potentialsof Musielak-Orlicz functions defined in a bounded metric measure space. As a corollary we obtain Trudinger's inequality for double phase functionals, whereandsatisfy log-Hölder conditions andis non-negative, bounded and Hölder continuous.

Trudinger's inequality for Riesz potentials on Musielak-Orlicz spaces over metric measure spaces