Some inclusion theorems for Orlicz and Musielak-Orlicz type spaces (Q1913395)

The integral operators of type \[ (Kf)(t)= \int_a^b K(t,s)f(s)ds \] are studied where \(]a,b[\) is a bounded interval on \(\mathbb{R}^+\), \(K\) is a (not necessarily homogeneous) kernel, and \(f\) is a function from Orlicz or Musielak-Orlicz type space \(L^\varphi(]a,b[)\). Some estimates are obtained that imply the property \[ f\in L^\varphi(]a,b[) \Rightarrow Kf\in L^\varphi (]a,b]). \] An interesting (strict in general) inclusion \[ L^\varphi(]a,b[)\subset \bigcap_{\alpha\in ]0,1[} L^{\varphi,\alpha} (]a,b[) \] is given where \(L^{\varphi,\alpha}\) denotes the fractional Orlicz space of order \(\alpha\in [0,1]\) which is defined using a fractional derivative.