Teresa Signes

New classes of rearrangement-invariant spaces appearing in extreme cases of weak interpolation

We study weak type interpolation for ultrasymmetric spaces L?,E i.e., having the norm ??(t)f*(t)?E˜, where ?(t) is any quasiconcave function and E˜ is arbitrary rearrangement-invariant space with respect to the measure dt/t. When spaces L?,E are not “too close” to the endpoint spaces of interpolation (in the sense of Boyd), the optimal interpolation theorem was stated in [13]. The case of “too close” spaces was studied in [15] with results which are optimal, but only among ultrasymmetric spaces. In this paper we find better interpolation results, involving new types of rearrangement-invariant spaces, A?,b,E and B?,b,E, which are described and investigated in detail.

Remarks on (Q, P, Y)-Summing Operators

An operator T ∈ ΠY q,p, (Xˇ⊗ Y, Z) if there exists a constant C > 0 such that, for any finite sequence u 1 , u 2 ,..., u N inX ⊗ Y, we have (N k =1σ||T(uk)||q Z) 1/q ≤ C sup x ∗ ∈ B X ∗{(Nσk =1||uk (x ∗)||p1/p}. It is shown that if T ∈ ΠY p (Xˇ ⊗ Y, Z) and S ∈ Π s, t(Z, W) then the operator ST ∈ Π Y r, q (X ⊗ Y, W) where 1/r = 1/p + 1/s and 1/q = 1/p + 1/t.