Grand Sobolev spaces and their applications in geometric function theory and PDEs

Abstract

Function spaces that are slightly larger than the Lebesgue Lp(Ω) spaces (even larger than the Marcinkiewicz Lp,∞(Ω) spaces) have been introduced by Iwaniec and Sbordone [Arch. Ration. Mech. Anal. 119 (1992), 129–143] in connection with integrability properties of the Jacobian. These are the grand Lebesgue spaces Lp)(Ω). In this survey we collect a number of results which prove that these spaces are useful in various classical settings of geometric function theory and partial differential equations (PDEs).