Under appropriate assumptions on the $N(\Omega)$-fucntion, locally uniform Morrey estimate is presented in Musielak-Orlicz-Sobolev space. The include a new increasing condition $x$-derivative of Young complementary function $N(\Omega)$-fucntion. conclusion applies to several important nonlinear examples frequently appeared mathematical literature.

It is proved that a Musielak-Orlicz space LΦ of real valued functions which is isometric to a Hilbert space coincides with L2 up to a weight, that is Φ(u,t)= c(t)u2. Moreover it is shown that any surjective isometry between LΦ and L∞ is a weighted composition operator and a criterion for LΦ to be isometric to L∞ is presented. Isometries in complex function spaces have been studied successfully ...

In this paper we give a survey about recent versions of Korovkin-type theorems for modular function spaces, class which includes $L^p$, Orlicz, Musielak-Orlicz spaces and many others. We consider various kinds convergence, using certain summability processes, like triangular matrix statistical filter convergence (which are generalizations the convergence). Finally, wwe an abstract axiomatic pre...

We consider a parabolic PDE with Dirichlet boundary condition and monotone operator $A$ non-standard growth controlled by an $N$-function depending on time spatial variable. do not assume continuity in for the $N$-function. Using additional regularization effect coming from equation, we establish existence of weak solutions particular case isotropic $N$-function, also prove their uniqueness. Th...