Extrapolation theory: new results and applications

Abstract

Extending earlier work by Jawerth and Milman, we develop in detail (p) and (p) methods of extrapolation. As an application we prove general forms of Yano's extrapolation theorem. Applications to logarithmic Sobolev inequalities, integrability of maps of finite distortion and logarithmic Sobolev spaces are given.

Notes on limits of Sobolev spaces and the continuity of interpolation scales

“…[19]) and the problem of computing the distance between interpolation spaces in a given scale (cf. [23]). …”

“…One intriguing question here is if there is a suitable normalization of the Rochberg-Weiss Ω operators (cf. [12], [23], and the references therein) that gives nontrivial commutator results at the end points.…”

“…For example, the rate of decay s −α leads to Dini-type conditions with logarithmic weights. We refer to [19] and [23] for descriptions of the spaces that will then appear as limiting spaces.…”

Grand Orlicz spaces and global integrability of the Jacobian

“…Our investigation is closely related to [7], [4], where the second and the third authors studied the grand and small Lebesgue spaces and their analogs. The main difference with [7] is that now we do not use the general interpolation-extrapolation theory, although the technique from [13] is applied. The reason for this choice is that the real interpolation of the Orlicz spaces requires too strong conditions on the Orlicz functions.…”

Qaisar Mehmood. Optimal Embeddings of Generalized Sobolev Spaces in Hölder–Zygmund Spaces

“…In some cases it can be simplified. To this end, we apply the Σ q −method of extrapolation [ 3 ] from the supercritical case. As a byproduct, we also characterize the embedding W k E ֒→ C j , j < k, where C j consists of all functions with bounded and uniformly continuous derivatives up to order j. Namely, this is equivalent to the embedding E ֒→ L n/(k−j),1 if k ≤ n. The embedding W n+j E ֒→ C j is always true since W n E ֒→ W n 1 ֒→ C 0 .…”