Grand and Small Lebesgue Spaces and Their Analogs

Abstract

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.

Grand Orlicz spaces and global integrability of the Jacobian

“…They have applications to some boundary value problems, see [17], [6]. 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.…”

“…For example, if (t) = t p , h(t) = 1, N(σ ) = σ , and μ(X) = 1, then we get the grand Lebesgue space L p) ([12], [7]). In this case (see Proposition 2.6 below)…”

“…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.…”

“…On the other hand, we use Lorentz-Orlicz classes, the quasinorm in which has the same structure as that given by the K-functional in the real interpolation. Therefore a direct approach is possible, which enable us to give characterizations of the grand Lorentz-Orlicz spaces and the grand Orlicz spaces (Theorem 2.4 and Proposition 2.6), similar to those for the grand Lebesgue spaces in [7].…”

Extrapolation properties of closed operator ideals

“…The following extrapolation spaces are considered in [17,13,15,6,5] and [8] among other papers. Subsequently, given positive functions f, g defined on an interval I, we write f (t) ∼ g(t) if there are constants c 1 , c 2 …”