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

Tom 170 / 2005

N. Krugljak, Y. Sagher, P. Shvartsman Studia Mathematica 170 (2005), 173-201 MSC: Primary 46B70. DOI: 10.4064/sm170-2-4


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.


  • N. KrugljakDepartment of Mathematics
    Luleå University of Technology
    SE-971 87 Luleå, Sweden
  • Y. SagherDepartment of Mathematical Sciences
    Florida Atlantic University
    Boca Raton, FL 33431-0991, U.S.A.
  • P. ShvartsmanDepartment of Mathematics
    Technion–Israel Institute of Technology
    32000 Haifa, Israel

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