The Quasinormed Neumann–schatten Ideals and Embedding Theorems for the Generalized Lions–peetre Spaces of Means

For the spaces φ(X0, X1)p0,p1 , which generalize the spaces of means introduced by Lions and Peetre to the case of functional parameters, necessary and sufficient conditions are found for embedding when all parameters (the function φ and the numbers 1 ≤ p0, p1 ≤ ∞) vary. The proof involves a description of generalized Lions–Peetre spaces in terms of orbits and co-orbits of von Neumann–Schatten ideals (including quasinormed ideals). The first interpolation theorem for linear operators, proved by M. Riesz in [17], states practically that if a bounded linear operator T sends simultaneously Lp0 to Lq0 and Lp1 to Lq1 , where p0 ≤ q0 and p1 ≤ q1, then T sends Lp, where p0 < p < p1, to the space Lq such that the partition of the interval [1/q0, 1/q1] by 1/q is similar to the partition of [1/p0, 1/p1] by 1/p. We also have the corresponding multiplicative estimate for the norm of the operator T we interpolate. Several decades later it became clear that this theorem can be refined considerably. It turned out that T actually sends Lp to the Lorentz space Lq,p, which is smaller than Lq if p0 < q0 or p1 < q1. Moreover, the space Lq,p cannot be reduced if we consider all operators that map {Lp0 , Lp1} to {Lq0 , Lq1}. The latter result belongs to Dikarev and Matsaev [4]. Thus, the problem of existence of the smallest target Banach space arose, for a given domain space E and all bounded linear operators that map one couple {E0, E1} of Banach spaces to another couple {F0, F1} of Banach spaces. It turned out that such a space exists indeed; Aronszajn and Gagliardo [1] called this space the interpolation orbit of E with respect to bounded linear operators mapping {E0, E1} to {F0, F1}, denoting it by Orb(E, {E0, E1} → {F0, F1}). The Riesz theorem was extended by Thorin to arbitrary Lp; i.e., the conditions p0 ≤ q0 and p1 ≤ q1 were dropped. The sharp theorem in the general case appeared only in the 1980s. From the very beginning, that theorem was stated in the framework of the Lorentz spaces Lp,q. These spaces reflect the subtler behavior of measurable functions, and the family of all Lorentz spaces contains all Lp. It was proved in [11] that if a linear operator T sends Lp0,s0 to Lq0,t0 , and Lp1,s1 to Lq1,t1 for some 1 ≤ p0, p1, q0, q1, s0, s1, t0, t1 ≤ ∞, then T sends Lp,s to Lq,t, where 1/p = (1− θ)/p0 + θ/p1, 1/q = (1− θ)/q0 + θ/q1, 1/t = 1/s+ (1− θ)(1/t0 − 1/s0)+ + θ(1/t1 − 1/s1)+, for all 0 < θ < 1. (As usual x+ = max(x, 0).) 2010 Mathematics Subject Classification. Primary 46M35, 46B70.