Marcinkiewicz interpolation theorems for Orlicz and Lorentz gamma spaces. (English) Zbl 1305.46020

Kerman, Ron; Phipps, Colin; Pick, Lubosš

Publ. Mat., Barc. 58, No. 1, 3-30 (2014).

Summary: Fix the indices \(\alpha\) and \(\beta\), \(1<\alpha<\beta<\infty\), and suppose \(\varrho\) is an Orlicz gauge or Lorentz gamma norm on the real-valued functions on a set \(X\) which are measurable with respect to a \(\sigma\)-finite measure \(\mu\) on it. Set \[ M(\gamma,X):=\{f: X\to\mathbb R \text{ with } \sup_{\lambda>0}\lambda \mu(\{x\in X: |f(x)|>\lambda\})^{\frac1{\gamma}}<\infty\}, \] \(\gamma=\alpha,\beta\). In this paper, we obtain, as a special case, simple criteria to guarantee that a linear operator \(T\) satisfies \(T: L_{\varrho}(X)\to L_{\varrho}(X)\), whenever \(T: M(\alpha,X)\to M(\alpha, X)\) and \(T: M(\beta,X)\to M(\beta, X)\).

MSC:

46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Keywords:

supremum operators; Sobolev imbeddings; optimality; rearrangement-invariant norms