Weak-type operators and the strong fundamental lemma of real interpolation theory

N. Krugljak ¹   ; Y. Sagher ²   ; P. Shvartsman ³

¹ Department of Mathematics Luleå University of Technology SE-971 87 Luleå, Sweden
² Department of Mathematical Sciences Florida Atlantic University Boca Raton, FL 33431-0991, U.S.A.
³ Department of Mathematics Technion–Israel Institute of Technology 32000 Haifa, Israel

Studia Mathematica, Tome 170 (2005) no. 2, pp. 173-201

Résumé

We prove an interpolation theorem for weak-type operators. This is closely related to interpolation between weak-type classes. Weak-type classes at the ends of interpolation scales play a similar role to that played by ${\rm BMO}$ with respect to the $L^{p}$ interpolation scale. We also clarify the roles of some of the parameters appearing in the definition of the weak-type classes. The interpolation theorem follows from a $K$-functional inequality for the operators, involving the Calderón operator. The inequality was inspired by a $K$-$J$ inequality approach developed by Jawerth and Milman. We show that the use of the Calderón operator is necessary. We use a new version of the strong fundamental lemma of interpolation theory that does not require the interpolation couple to be mutually closed.

Détail

DOI : 10.4064/sm170-2-4

Keywords: prove interpolation theorem weak type operators closely related interpolation between weak type classes weak type classes ends interpolation scales play similar role played bmo respect interpolation scale clarify roles parameters appearing definition weak type classes interpolation theorem follows k functional inequality operators involving calder operator inequality inspired k j inequality approach developed jawerth milman calder operator necessary version strong fundamental lemma interpolation theory does require interpolation couple mutually closed