We give the following, equivalent, explicit expressions for the norms of the small and grand Lebesgue spaces, which depend only on the non-decreasing rearrangement (we assume here that the underlying measure space has measure 1): f L (p ≈ [f * (s)] p ds 1 p (1 < p < ∞). Similar results are proved for the generalized small and grand spaces.

No 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.

Sharp reiteration theorems for the K-interpolation method in limiting cases are proved using two-sided estimates of the K-functional. As an application, sharp mapping properties of the Riesz potential are derived in a limiting case.

No abstract

No abstract

No abstract

We compute the limits of higher-order Besov norms and derive the sharp constants for certain forms of the Sobolev embedding theorem. Our results extend to higher-order spaces the recent work by Brézis-Bourgain-Mironescu and Maz'ya-Shaposhnikova. The interpolation methods we develop are of interest on their own and could have applications to related inequalities.