Interpolation of linear operators in Orlicz spaces (Q5903788)

In this short note there are considered Orlicz spaces \(L^*_{\phi}(\Omega,\mu)\) defined for a given N-function \(\phi\) \((\phi \approx u^ p,0<p<\infty)\) and in the case when measure \(\mu\) is finite. The main result of the note is a theorem. This theorem states that if \(\phi,\psi,\phi_ i,\psi_ i\) \((i=0,1)\) are N-functions such that \(\psi (u)\approx u^ r\), \(\phi (u)\approx u^ s\), \(\psi_ i(u)\approx u^{q_ i}\), \(\phi_ i(u)\approx u^{p_ i}\) \((i=0,1)\) where \(1/s=\theta /p_ 1+(1-\theta)/p_ 0\), \(1/r=\theta /q_ 1+(1- \theta)/q_ 0\), \(0<\theta <1\) and if \(T:L^*_{\phi_ i}\to L^*_{\psi_ i}\), \((i=0,1)\), then the operator \(T:L^*_{\phi}\to L^*_{\psi}\) is continuous. The note does not contain proofs and remarks on \(\epsilon\) in the definition of \(\approx\).