Trudinger’s Inequality for Double Phase Functionals with Variable Exponents

Our aim in this paper is to establish Trudinger’s inequality on Musielak-Orlicz-Morrey spaces L Φ, κ ( G ) under conditions on Φ which are essentially weaker than those considered in a former paper. As an application and example, we show Trudinger’s inequality for double phase functionals Φ( x, t ) = t p ( x ) + a ( x ) t q ( x ) , where p (·) and q (·) satisfy log-Hölder conditions and a (·) is nonnegative, bounded and Hölder continuous.