We give a complete characterization of limiting interpolation spaces for the real method of interpolation using extrapolation theory. For this purpose the usual tools (e.g., Boyd indices or the boundedness of Hardy type operators) are not appropriate. Instead, our characterization hinges upon the boundedness of some simple operators (e.g. f → f (t 2 )/t, or f → f (t 1/2 )) acting on the underlying lattices that are used to control the K-and J-functionals. Reiteration formulae, extending Holmstedt's classical reiteration theorem to limiting spaces, are also proved and characterized in this fashion. The resulting theory gives a unified roof to a large body of literature that, using ad-hoc methods, had covered only special cases of the results obtained here. Applications to Matsaev ideals, Grand Lebesgue spaces, Bourgain-Brezis-Mironescu-Maz'ya-Shaposhnikova limits, as well as a new vector valued extrapolation theorems, are provided.

We introduce the notion of a rearrangement invariant extrapolation space on [0, 1]. We obtain some sufficient conditions (also necessary in some cases) for the Marcinkiewicz and Lorentz spaces to be extrapolation spaces. We generalize and improve the previous results, which enables us to determine the possible limits of such description of spaces and thereby to establish assertions of the Yano-type theorem. In particular, we present some examples of the spaces "close" to L ∞ in a sense but lacking this description.

We introduce a new class of rearrangement invariant spaces on the segment [0, 1] which contains the most common extrapolation spaces with respect to the L p -scale as p → ∞. We characterize the class and demonstrate under certain conditions that the Peetre K -functional has extrapolatory description in the couple (E, L ∞ ) if and only if E belongs to the class. By way of application, we establish a new extrapolation theorem for the bounded operators in L p .

We reformulate, modify and extend a comparison criteria of L p norms obtained by Nazarov-Podkorytov and place it in the general setting of interpolation theory and majorization theory. In particular, we give norm comparison criteria for general scales of interpolation spaces, including noncommutative L p and Lorentz spaces. As an application, we extend the classical Ball's integral inequality, which lies at the basis of his famous result on sections of the n−dimensional unit cube.