Last modified: 2023-05-15

#### Abstract

A generalized Orlicz space, also known as a Musielak--Orlicz space, consist of those measurable functions that satisfies

\[

\int_\Omega \phi(x, \lambda |f(x)|) \, dx < \infty

\]

for some $\lambda >0$. Here $\phi: \Omega \times [0, \infty] \to [0, \infty]$ satisfies some natural assumptions. As special cases these function spaces include classical weighted Lebesgue spaces $\phi(x, t):= a(x) t^p$, Orlicz spaces $\phi(x, t):= \phi(t)$, variable exponent Lebesgue spaces $\phi(x, t):= t^{p(x)}$ and double phase -spaces $\phi(x, t):= t^p + a(x) t^q$. We discuss about properties of these spaces and their applications to the calculus of variations. The speak is mainly based on recent book "Orlicz spaces and Generalized Orlicz spaces", Lecture Notes in Math 2236 Springer, by Peter Hästö and the author.