Inclusion indices of function spaces and applications

Abstract

We investigate inclusion indices for general function spaces, not necessarily symmetric. Using them, we estimate the grade of proximity between two spaces E ↪→F when we have certain information on the inclusion. The results are based on ideas from interpolation theory. 0. Introduction Let E and F be symmetric (that is, rearrangement invariant) spaces. A bounded linear operator T ∈L(E,F ) is said to be disjointly strictly singular (or a DSS operator) if there is no disjoint sequence of non-null functions {fn} in E so that the restriction of T to the subspace [fn] spanned by the functions {fn} is an isomorphism (see [10]). For example, if 1 q 0, the Peetre K-functional is the norm on A0 +A1 given by K(t, a) = K(t, a;A0, A1) = inf{‖a0‖A0 + t‖a1‖A1 : a = a0 + a1, aj ∈ Aj}. Any two of these norms are equivalent. We put K(1, ·) = ‖ · ‖A0+A1 . The J-functional is the norm on A0 A1 defined by J(t, a) = J(t, a;A0, A1) = max{‖a‖A0 , t‖a‖A1}. Again, any two of them are equivalent. Put J(1, ·) = ‖ · ‖A0 A1 . The Kand the Jfunctional are related by duality. More precisely, if A0 A1 is dense in A0 and in A1 then (see [3, theorem 2·7·1]) (A0 +A1) = A0 A ∗ 1 and 1 t J(t, f ;A1 , A ∗ 0) = sup a∈A0+A1 |f (a)| K(t, a;A0, A1) . (1·1) We say that a Banach space A is an intermediate space with respect to the Banach couple (A0, A1) if A0 A1 ↪→A ↪→A0 +A1, where ↪→means continuous inclusion. Associated to A we have the functions ρA(t) = ρA(t;A0, A1) = inf{J(t, a;A0, A1) : a ∈ A0 A1, ‖a‖A = 1} and ψA(t) = ψA(t;A0, A1) = sup{K(t, a;A0, A1) : a ∈ A, ‖a‖A = 1} (see [4]). The functions ρA and ψA are similar to functions which have been studied by Dmitriev [8] and Pustylnik [15]. Note that ρA and ψA are strictly positive and non-decreasing, and that ρA(t)/t and ψA(t)/t are non-increasing. Hence, if we put ρA(0) =ψA(0) = 0 then we get two quasiconcave functions. Here we shall only work with the Banach couples (L1(Ω, μ), L∞(Ω, μ)), where (Ω, μ) is a σ-finite measure space. We shall pay special attention to the cases Ω = [0, 1] with μ the Lebesgue measure and Ω=N with μ({n}) = 1 for each n∈N. The intermediate Inclusion indices of function spaces and applications 667 spaces will be Banach spaces E of measurable functions on Ω. Sometimes we shall assume that they are lattices, that is, whenever |g(x)| |f (x)|μ−a.e. with f ∈ E and g measurable, then g ∈E and ‖g‖E ‖f‖E. We also recall that a Banach lattice E is termed a symmetric space if whenever f ∈E and g is equimeasurable with f , then g ∈ E and ‖f‖E = ‖g‖E . IfE is symmetric then the function φE(t) = ‖χD‖E where D⊂Ω with μ(D) = t is well-defined and it is called the fundamental function of E. It turns out that φE(t) = t ρE(t;L1(Ω), L∞(Ω)) = t ψE(t;L1(Ω), L∞(Ω)) . 2. Function spaces and inclusion indices Let E be a Banach space of measurable functions on [0, 1] with L∞[0, 1] ↪→ E ↪→L1[0, 1]. As in the symmetric case, we define the inclusion indices of E by δE = sup{p 1 : E ↪→ Lp[0, 1]}, γE = inf{p ∞ : Lp[0, 1] ↪→ E}. The next results show that in our general case the indices are related to the functions ψE and ρE. Theorem 2·1. Let E be a Banach space of measurable functions on [0, 1] with L∞[0, 1] ↪→E ↪→L1[0, 1]. Then δE = lim inf t→0 log t log(t/ψE(t)) . Proof. Wemay assume, without loss of generality, that the norm of the embedding L∞[0, 1] ↪→E is 1. We claim that t/ψE(t) 1 for any t ∈ (0, 1). (2·1) Indeed, since‖χ[0,t]‖E ‖χ[0,t]‖L∞=1and since theK-functional on (L1[0, 1], L∞[0, 1]) is given by K(t, f ) = ∫ t 0 f ∗(s) ds where f∗(s) = inf{σ > 0 :m{x : |f (x)|>σ} s} (see [3, theorem 5·2·1]), we have ψE(t) K ( t, χ[0,t] ) ‖χ[0,t]‖E ∫ t